This is a relatively straight forward question and answer. Yes, Zen, or Chan/Thien/Seon, is Buddhism. r/zen is a problematic community which broadly attempts to deemphasize the religious aspects of Buddhism, and is a part of a modern neo-colonial movement to secularize Buddhism from its long-standing traditions and ideas.

Archeological and historical record points to Buddhism reaching China around the Han Dynasty, though it begins to take off around the 4th-5th century, and subsequently flourishing by the Tang. Importantly, Chinese Buddhism is relatively non-sectarian, meaning multiple different traditions and schools intermingle and do not have clear distinctions separating them from interacting and learning from one another. A typical Chinese Buddhist temple may have monks who meditate according to Chan principals, while reciting Pureland sutras. Chan narratives state that Bodhidharma in the 5th century brought the school to China. He was the twenty-eighth Indian patriarch of Buddhism, and would then become the First Patriarch of Chan.

Chan emphasizes meditation, however, meditation is common to most if not all Buddhist traditions. Meditation itself is a fairly specialized and niche activity, mostly done by monastics and trained laypersons. It's a part of various other Buddhist practices, such as reading, studying, and chanting liturgy, doing prostrations, circumambulations, prayer, rituals, etc. Chan is itself heavily influenced by the Huayan school, which was popular in the Tang period alongside contemporary schools such as Tiantai. These schools had major political support from the Chinese emperors, which helped to fund institutions and disseminate the teachings. While Tiantai and Huayan may heavily value and rely on scriptures, Chan advocates more for practice, experience, and meditation over scholasticism and study. Chan would also have strong political allegiances, and the state would require monastics, including those from the Chan school, to be versed in the sutras and follow the Vinaya to fully qualify for novice and full monastic ordinations.

Central to Buddhist philosophy is the concept of anatta or anatman, non-self. Things do not exist separately as its own individual thing, it is reliant upon other things in order to exist. All phenomena are interdependent, existing through constantly changing things. Huayan philosophy centers around a non-dualistic view of the world, expanding on interdependence to become interpenetrating. In other words, "everything" is just "one thing".

Consider a building. The building is made up of walls, pillars, rafters, a roof. But remove those walls, pillars, rafters, a roof, is it still a building? No, because without the rafters, the building cannot stand, and so it no longer functions as a building. But conversely, what are the walls, pillars, rafters, and roof? What makes a rafter a rafter? Well, by being the rafter of a building, it's a rafter, otherwise by itself it's nothing but a piece of wood. They make up the building, so they "are" the building. Remove it from the building, and it is no longer the building, as it is no longer a rafter. Things are nothing but the sum of their parts. Therefore, each part is essential to its identity. The rafter is the building. The building is the rafter. The rafter is the universe. The universe is the rafter. Everything contains within it everything. All is one, one is all. That's the non-dualistic worldview of Huayan.

Take that to Chan Buddhism. Everything has within itself Buddha-nature, because everything is essentially the same thing, so how can it now contain within itself Buddha-nature? The Platform Sutra discusses gradual and sudden enlightenment, but everything is all one practice because everything is the same, there should be no distinguishing of practice, that'd be dualistic not monistic. Every act is a Buddha-like act, let everything be Buddha-nature, we just need to become aware and enlightened to that fact.

The Platform Sutra is an important text to the Chan tradition, written by the sixth patriarch Huineng. Lineage is paramount in Chan Buddhism. It effectively legitimizes the tradition as being Buddhist, as being true to Sakyamuni/Gautama Buddha. By tracing a supposedly uninterrupted lineage of masters and students, Chan creates a narrative that it is upholding and continuing the teachings of Buddhism transmitted straight from the founder of the religion. A common lineage may be identified as Sakyamuni ->Kasyapa -> Ananda -> Bodhidharma -> Huike -> Sengcan -> Daoxin... which then splits off to different schools, such as the Oxhead School, Northern and Southern schools from Hongren and Huineng, and from that to the Five Houses, and Linji and Caodong, or Rinzai and Soto, popular in Japan. Even though the historical veracity of these lineages and timelines may not be completely accurate, what's important for Chan Buddhists is that they themselves consider it accurate, real, and trustworthy. This lineage stands out for the Chan Buddhists as it allows them to claim it is the true essence of Buddhism, that it represents Gautama's teachings in the highest form. This was important when countering rival movements such as the Tiantai school, which advocated for the Lotus Sutra as the paramount text and had strong political backing. Though again, there was never such a stark divide as would develop in Japan that cross-school practice was prohibited. Whatever the case, the very fundamental identity of Chan is that it is a special, true, and realized form of Buddhism that is optimal for guiding followers to enlightenment and universal salvation.

The Platform Sutra extensively discusses Dharma, Buddha nature, the importance of meditation, and various other key Buddhist concepts and practices. Here is one exercpt:

"The Master Huineng called, saying: "Good friends, bodhi and prajna are from the outset possessed by men of this world themselves. It is just because the mind is deluded that men cannot attain awakening to themselves. They must seek a good teacher to show them how to see into their own natures. Good friends, if you meet awakening, Buddha-wisdom will be achieved."

"Good friends, my teaching of the Dharma takes meditation and wisdom as its basis. Never under any circumstances say mistakenly that meditation and wisdom are different; they are a unity, not two things. Meditation itself is the substance of wisdom; wisdom itself is the function of meditation. At the very moment when there is wisdom, then meditation exists in wisdom; at the very moment when there is meditation, then wisdom exists in meditation. Good friends, this means that meditation and wisdom are alike. Students, be careful not to say that meditation gives rise to wisdom, or that wisdom gives rise to meditation, or that meditation and wisdom are different from each other. To hold this view implies that things have duality-if good is spoken while the mind is not good, meditation and wisdom will not be alike. If mind and speech are both good, then the internal and the external are the same and meditation and wisdom are alike. The practice of self-awakening does not lie in verbal arguments. If you argue which comes first, meditation or wisdom, you are deluded people. You won't be able to settle the argument and instead will cling to objectivethings, and will never escape from the four states of phenomena.

"The samadhi of oneness is straightforward mind at all times, walking, staying, sitting, and lying. The Vimalakirti Sutra says: 'Straightforward mind is the place of practice; straightforward mind is the Pure Land.' Do not with a dishonest mind speak of the straightforwardness of the Dharma. If while speaking of the samadhi of oneness, you fail to practice straightforward mind, you will not be disciples of the Buddha. Only practicing straightforward mind, and in all things having no attachments whatsoever, is called the samadhi of oneness. The deluded man clings to the characteristics of things, adheres to the samadhi of oneness, thinks that straightforward mind is sitting without moving and casting aside delusions without letting things arise in the mind. This he considers to be the samadhi of oneness.

Much of the Platform Sutra discusses the totality of practice, in that meditation cannot just be something done while seated and quiet, but that practice is everything, coming off of the Huayan ideas of non-dualism. If everything contains within it Buddha-nature, then every action, "meditation" or "not meditation", is an act of Buddha-nature. The Platform Sutra also discusses things such as the flaws of the Northern School from Shenxiu, and the primacy of Huineng's teachings. It contends that Chan Buddhism is the best path of Buddhism, and that other schools are deluded or only partially correct.

[continued]

In addition to what u/huianxin wrote, which I think is excellent and with which I agree entirely, let me add this: It is...puzzling that one could make a serious study of the Zen commentarial literature and come away with the impression that Zen did not see itself as a branch of mainstream Mahayana Buddhism.

(n.b., for simplicity's sake, I'm either going to use English translations of Chinese words or titles where possible--thus, "Gateless Gate" instead of "Wumenguan"--or, where the Japanese is more familiar, I'll use that--thus, "koan" instead of "kung-an" or "public case"; generally "Zen" instead of "Chan," except where the distinction is important. Proper names I'll leave as they are in the text.)

It is my understanding that r/zen views as "legitimate Zen" only those Chinese Chan teachers from the Tang and Song periods: so, no modern Chan teachers, no Japanese Zen, no Vietnamese Thien, no Korean Seon--and certainly no westerners (excepting themselves). So for the sake of argument, let's set aside the commentarial literature from those later traditions and focus on that from that window of legitimacy... and still we find that literature to be entirely shot through with Buddhist iconography, stories from Buddhist scriptures, and Buddhist themes.

For example, Zen's legendary founding comes from an encounter between the historical Buddha, Sakyamuni, and his disciple Mahakasyapa. This founding story is recounted as Case 6 of the Gateless Gate (one of the major koan collections, compiled in 1228 by master Wumen Huikai):

At a gathering on Vulture Peak, the World-Honored One held up a flower and showed it to the assembly. At that moment, everyone in the assembly was silent except Mahakasyapa, who broke into a smile.

The World-Honored One said, "I have the treasury of the true dharma eye, the wondrous mind of nirvana, the true form of no-form, the subtle and wondrous gate to the dharma, the special transmission outside of scriptural teachings not established on words and language. I now entrust it to Mahakasyapa."

In other words, Zen's founding legitimacy hinges on the transmission from Sakyamuni Buddha to Mahakasyapa. It would be odd for Zen simultaneously to rely on Sakyamuni for legitimacy and to disclaim his teaching. Nor is this the only time that the historical Buddha or his contemporaries serve as protagonists in these stories. For example, the first case in the Book of Serenity (a koan collection comprising 100 koans drafted by Hongzhi Zhengjue in the 12th Century and compiled with commentary about a hundred years later by Wansong Xingxiu) is titled "The World Honored One Ascends the Seat":

One day the World Honored One ascended the seat.

Manjusri [another of the Buddha's chief disciples, and famous in Mahayana Buddhism as the Bodhisattva of wisdom; the archetype of wisdom personified] struck the gavel and said, "Clearly observe the Dharma of the King of Dharma; the Dharma of the King of Dharma is thus."

The World Honored One then got down from the seat.

Or take Case 22 of the Gateless Gate, which tells of an encounter between Mahakasyapa and Ananda, the Buddha's chief attendant:

Ananda asked Mahakasyapa, "Besides the golden robe, what else did the Buddha transmit to you?"

Mahakasyapa yelled, "Ananda!" Ananda replied, "Yes?" Mahakasyapa said, "Take down the temple flagpole in the front gate."

For context, one would need not only to know that Mahakasyapa received the transmission of Sakyamuni's Dharma, but to know Ananda's story: Ananda was thought to be the great memorizer of Sakyamuni's words and among the most learned of Sakyamuni's disciples, but famously did not realize awakening during Sakyamuni's lifetime. So the context here is that Mahakasyapa had achieved awakening, Ananda had not, and Ananda was trying to figure it all out. In his great compassion, Mahakasyapa was trying to help Ananda along. To fully understand this koan, therefore, one would need to know both Zen's founding myth, Ananda's struggle to achieve liberation, and the relative timeline of the two. One would, in other words, need a foundation in basic Buddhist history.

There are other koans that refer to the Buddha and his disciples (e.g., Case 4 of the Book of Serenity; Case 32 of the Gateless Gate; Case 65 of the Blue Cliff Record). And beyond these references to Buddhist personages, the koans frequently quote Buddhist scriptures. For example, Case 48 of the Book of Serenity is titled "Vimalakirti's 'Nonduality'":

Vimalakirti asked Manjusri, "What is a Bodhisattva's method of entering nonduality?"

Manjusri said, "According to my mind, in all things, no speech, no explanation, no direction and no representation, leaving behind all questions and answers--this is the method of entering nonduality."

Then Manjusri asked Vimalakirti, "We have each spoken. Now you should say, good man, what is a Bodhisattva's method of entry into nonduality?"

Vimalakirti was silent.

As recounted in the koan, this encounter is a slightly condensed version of Chapter 9 of the Vimalakirti Sutra. Here, for example, is how the story is told in Robert Thurman's translation. In the Sutra, the bodhisattva Vimalakirti has gathered to him a host of Sakyamuni's most realized disciples and bodhisattvas, and has asked them all to explain the Dharma-door of nonduality. Each bodhisattva provides their answer; all are facially correct but ultimately miss the mark. Then they turn to Manjusri, the wisest of them all:

When the bodhisattvas had given their explanations, they all addressed the crown prince Manjusri: "Manjusri, what is the bodhisattva's entrance into nonduality?"

Manjusri replied, "Good sirs, you have all spoken well. Nevertheless, all your explanations are themselves dualistic. To know no one teaching, to express nothing, to say nothing, to explain nothing, to announce nothing, to indicate nothing, and to designate nothing--that is the entrance into nonduality."

Then, the crown prince Manjusri said to the Licchavi Vimalakirti [Licchavi was the name of Vimalakirti's clan], "We have all given our own teachings, noble sir. Now, may you elucidate the teaching of the entrance into the principle of nonduality!"

Thereupon, the Licchavi Vimalakirti kept his silence, saying nothing at all.

The crown prince Manjusri applauded the Licchavi Vimalakirti: "Excellent! Excellent, noble sir! This is indeed the entrance into the nonduality of the bodhisattvas. Here there is no use for syllables, sounds, and ideas."

The koan does not differ materially from the Sutra.

Nor was this the only koan which essentially recounts a Buddhist scripture. Case 45 of the Book of Serenity quotes one of the Prajanparamita Sutras (I think the Prajnaparamita in 8,000 Lines, but I'm not sure). Case 58 of the Book of Serenity quotes the Diamond Sutra, as does Case 97 of the Blue Cliff Record. Case 67 of the Book of Serenity quotes the Avatamsaka Sutra. Case 94 of the Blue Cliff Record quotes the Surangama Sutra. I'm sure there are some that I'm missing.

(continued)