This is a question that may appear on my PhD candidacy exams so I've been studying for it. Let's see how well I can do.

tl;dr: Buddhism was seen just as an alien curiosity at first. As it took root, a political concern arose about it being subversive or hostile to Chinese culture. This concern lasted for hundreds of years as it grew economically and politically powerful.

Buddhism was first transmitted to China as knowledge and technologies, not as a full-fledged monastic system. No one in ancient China was witness to a mysterious robed cult moving into their village and abruptly building a complex. Instead you have to imagine intellectuals showing up in the capital with an unheard-of new philosophy from a foreign land. And indeed, the initial reaction was that it was foreign and strange. The first person to translate Buddhist scriptures into Chinese was a foreigner called An Shigao who arrived in the Han capital of Luoyang in 148 CE. It has been suggested that he came from the Parthian Empire in Iran. We are not aware that anyone harassed him for his work, but China had a perfectly stable policy of governance, Confucianism, so the appeal of Buddhist doctrine was unclear at first. The concept of rebirth was inscrutable, and nirvana seemed irrelevant to Chinese existential concerns. We can see this in the choice of scriptures translated in this very early period in the Han capital:

If we examine the Han translations, we are confronted with a curious situation. Not many of the translations are concerned with the fundamental doctrines of Buddhism: the four truths, chain of causation, nirvana, nonself, transitoriness, and so forth. Rather, most of them are concerned with concentration practices and breath control, subjects of interest to the Taoists. It appears that the foreign missionaries did not choose these texts for presen­tation to the Chinese; rather, it was the Chinese followers, often influenced by Taoism, who determined the choice, since such texts were the ones that interested them. Or perhaps the foreign monks chose to translate only those texts because they were the ones of interest to the Chinese. (Kenneth Ch'en, B in C, p.49)

For the next few centuries, translators often operated on the principle that Chinese readers of the sutras had no knowledge of Buddhism, so would culturally translate Buddhist philosophical concepts into corresponding Daoist or otherwise Chinese concepts.

In 220 AD, the Han dynasty collapsed, and we have the famous Three Kingdoms period which most people curious about China have encountered, followed by the southern dynasties of Jin. During this time, many Han Chinese were killed in civil war, and the continent was reoccupied by foreigners, to wit: "in a document of 299 AD the number of foreigners settled in [Chang'an in east-central China] is estimated at half a million, i.e. about one half of the total population" (Zuercher, B Conquest of C, p.81). Simultaneously, court officials of the Three Kingdoms and various other literati, faced with the total dissolution of their claims to legitimacy and violence everywhere, were radically rethinking both Confucian political philosophy and basic ontology, a philosophical deconstruction called Dark Learning (Xuanxue 玄學). Daoism played a role in this conversation, but so did Buddhism. Basically, the literati stopped gatekeeping their philosophical ideas based on country of origin, and looked for existential truths and logical retorts in Indian sutras.

So you had the general population of the continent becoming much more diverse in ethnicity and confused in political allegiance, simultaneous with a rethinking of philosophy at the top. This is when Buddhist monasteries began flourishing and receiving tax benefits. However, almost immediately, some people began exploiting this new system to get tax exemptions--not necessarily from greed, sometimes out of poverty and overtaxation. So the backlash to Buddhism also began in this time of war and disunity. In 335, Shi Hu of Later Zhao, a short-lived kingdom ruled by ethnic foreigners, attempted to ban new monastic ordinations after he had trouble collecting taxes. One of his Confucian ministers pointed out that Buddhism was a foreign religion so it should be banned altogether. Shi Hu replied that he was a foreigner as well so he didn't see the problem. This was the reality of the political situation after the end of the Han.

From our enlightened modern perspective, Buddhism was now flourishing with thousands of temples and monks, so we could claim it was part of Chinese culture beginning at this point and be done with this question. However, neither Buddhists nor non-Buddhist Chinese saw it that way. The Buddhists continued to struggle with how to relate to China and India. Their understanding of Indian philosophy had large gaps and hundreds of lengthy sutras required retranslations in order to better understand what the Indians meant. Monks would meditate but not know how to judge their progress, which led to hallucinations in meditation being interpreted to judge one's level of enlightenment, akin to dream interpretations. They did not even have a sutra containing the rules they were supposed to follow in their monasteries, and in 337, a monk named Faxian walked all the way to India in order to find the missing monastic rules document.

Meanwhile, the fact that monks were supposed to abandon their blood relations was a cause of anxiety both inside and outside the monasteries, because of the centrality of filial piety to Confucianism. In fact, Indian Buddhists were also concerned with relating to their families, but Confucianism and the suspicion of foreign teachings made this breaking of ties more politically upsetting in China.

So there were huge problems still separating Buddhism from normative Chinese behavior, and this led to four attempts to exterminate Buddhism entirely by various dynasties between 446 and 955. This large range of 500 years includes the Tang dynasty, when many innovative Buddhist ideas took root; on either side of the Tang were persecutions. It is really only in the Song dynasty (960–1279) that Buddhism became established as a Chinese institution, about 1000 years after it was first introduced, and after this, the political establishment continued to tweak tax benefits and so on as the monasteries became quite powerful landholders throughout the empire.

This was good practice for me summarizing one field I'm supposed to specialize in, but I would greatly appreciate it if any Chinese history expert could offer comments on this answer.