the people spoke Chinese,

The people spoke a variety of Koreanic and/or Japonic languages. (Apart from the language of Silla, the ancestor of modern Korean, the other languages spoken in Korea at the time are not well-known, since they were not written languages. Many place names from that time have been preserved, and strongly suggest a close relationship with modern Korean, and possibly Japanese.)

wrote in Chinese,

Classical Chinese was used as a written language. By this time, spoken Chinese had diverged from the written language, and Classical Chinese existed as a written rather spoken language, even in China. The situation in Korea (and Japan) where Classical Chinese was used as a written language, while the spoken language was different, was not that different from the situation in China. There was more difference between the written and spoken languages, since Korean and Japanese are not Chinese languages (both absorbed much of their vocabulary from Chinese languages, but kept their non-Chinese grammars).

used Chinese religion

The most influential religion in 6th century Korea was indigenous, with probable roots in religious traditions shared with ancient Manchuria and probably also Siberia. However, by the 6th century, Korean religion had already become distinctive. It owed very little to Chinese religion. Such indigenous religion is the source of the modern Korean tradition of shamanism, and has influence some modern Korean religions such Cheondoism (which grew out of the Donghak Rebellion in the 19th century).

Buddhism had been in Korea for over a century. It was still to become widely popular (which occurred in the 7th century), and although there were Buddhist schools of thought that had developed in Korea, it was still largely a foreign religion. Buddhism had been present in China for longer, but the situation was similar: there were Chinese schools of thought in Buddhism, but it was still largely a foreign religion. Buddhism and Buddhist thought was partly brought to Korea by Chinese missionaries, partly by missionaries from Central or South Asia, and partly by Korean who travels to China or Central/South Asia to study. The Chinese influence was strong, and a significant part of Buddhist thought was shared across China and Korea (and in the future, also Japan), but Korean Buddhism was developing on its own path.

used Chinese ... government.

6th century Korean governments were not patterned on Chinese government structure. In Silla, the traditional (and very non-Chinese) caste system, the bone-rank system, dominated. The Confucianisation of Korea lay in the future, and it seems that dissatisfaction with the rigid bone-rank based hierarchy of Silla was partly responsible for that coming Confucianisation.

Why are they considered Korean and not Chinese?

Apart from the above points, the Korean peninsula remained independent, and not part of China. Tang China had fought Goguryeo, and lost. The Korean kingdoms were fighting each other in an attempt to become the dominant power in Korea (and Japan was also involved in the fighting). Tang and Silla allied to defeat Goguryeo because they had a common enemy, and divided the territory of Goguryeo between them after their victory. Tang China got its share of the spoils, and Silla got its share, and became the unifier of the Korean peninsula. Silla was an ally of Tang China, not a subject of Tang China. Silla's unification of Korea brought about uniform government and more uniform language, and was important in shaping the future of Korea.

It was under later Korean dynasties (Goryeo and Joseon) that Chinese influence grew. Goryeo adopted Chinese-style civil service examinations, which brought Confucian education into the government. Joseon reinforced this, with an official strongly Confucian philosophy of government - the best-known Korean Neo-Confucian scholars, Toe-gye (Yi Hwang) and Yul-gok (Yi I) wrote in the 16th century. In the 6th century, this lay in the future. Korea was strongly influenced by China over the next millennium, and became tributary to China, but remained independent, with its own language, religion, and culture. It is possible that if Korea had become a Chinese province, Koreans today would be considered Chinese (compare the case of Yunnan), but as the first didn't happen, it should be unsurprising that the second didn't happen as a consequence.

(The part of Goguryeo lying in modern Manchuria was lost to Korea when the Tang-Silla alliance conquered Goguryeo. Today, while ethnic Koreans live there, they moved there much more recently.)

No, and that for basically all reasons you can state, as none of them do apply.

As the other answer gives a good general overview of your statements and why they are problematic, i'd like to add a linguistic explanation as it appears to happen quite often that people mistake a written script for being a spoken language as well, as by that statement, all of Europe speak a Romance language.

Korean constitutes its own linguistic family as, there haven't been found any valid relatives to the Korean language even so for the phonologically and morphologically similar Japanese language. The only serious contenders for being related are as far, Paleosiberian and Tungusic languages, Altaic languages and Japonic as the most likely one. Hence, Japanese is the only one that is discussed seriously about whether its related or not and linguists are split on that one as for example Vovin argues that it isn't due to the lack of native vocabulary shared by both langs since related langs usually have some cognates as it in the case of Indo-European languages, like middle (engl.), midjis (gothic), medius (lat.), mésos (greek), maiδya (iran.), meža (rus.), medis (Middle Welsh), etc.

Korean and Japanese do have shared cognates, such as Island, "Seom" in Korean and "Soma" in Japanese, bear - "Geom" (k.), "Koma (j.) or if you go back in time the word for morning is in modern Korean "acchim" and is in Japanese "Asa", in Old Korean it was "Sasa" and in OJ it was "Soa".

So if they two are related, it has to be relatively far back, around as much as English and Hindi for example due to the low number of cognates. Which would be around 4 millenia BP.

And that is the case with a language that is the "closest" one linguistically. And if they are related they picked up words from neighbouring languages and evolved completely different vocabularies whereas their structure remained similar.

Japanese and Korean both have phonetic alphabets that match their respective languages though Japanese still used Chinese characters for writing and Korea used Chinese characters adapted to Korean pronounciation and sounds in the past.

But they didn't couldn't use Chinese script as it is, and the one they derived Hanja/Hyangchal and Hyangga from was from Old Chinese. So couldn't the Chinese really use Korean words written down in Chinese characters, wheras the Koreans changed the Chinese words in the way it could be properly read in the Korean language.

As for the religion. The native religions of Korea are very much alike all the other Northeast Asian native religions and share most of their features, such as Shamanism (Mudang) and Animism (Shindo), as well as general superstitions about ghosts (Ghisshin) and creatures (Shinkye). Shindo has a broad overlap to the Japanese Shinto, and may have the same origin.

Besides that Korean native religion also has Totems like the Chukchi and Na-Dene people in NA.

Korea very much had a feudal, fractal society and not a hierarchichal, imperial bureaucratic government until the early 15th century.