Well, it does not have an economy, so why would it have money?
Also, it doesn’t have politics and society in the conventional sense, but men are clearly subordinate to God. Christ is king, this is the way Christians think, so I am not sure this is a correct comparison.
The question of “should Christians strive for a classless society” is a complex one. Egalitarian ideals are very new compared to Christianity, but some Christians now think that in the “fallen world” authority is undesirable as it can be abused. This is not common though.
However, Marxism is an anti-religious ideology. Marxists both believe that religion will disappear after “the base” changes and it will become, ultimately, obsolete, and also have historically persecuted and enacted violence on Christians. So I am not surprised there are not many Marxist Christians.
That would be irrelevant even if it was true. We are not in the second century. It is a very controversial position either way.
Egalitarian values certainly did emerge out of Christianity, and there was a change in that direction even then, but they were not egalitarian in the modern sense.
Also, please be careful when generalising early Christianty, as it was a very diverse group of sects that hardly agreed on anything.
Early religious communities sometimes were very accepting, and women played a role as well, but they still existed in a very patriarchal culture, so you should not expect their women to be equal to men in society, and there were absolutely positions of authority.
They opposed the empire because initially, they were not perceived by anyone as a group distinct from Jews, which were very hostile to it. However, there were appeals made by powerful Christians later to be recognized as a non-threat to imperial power, and ultimately, they succeeded.
Even so, the Jews simply wanted independence, not equality. The idea of social equality did not even exist then. They were equal in Christ, not in society.
Christianity was not coopted by the empire, it conquered it.
The idea that early christianity was somehow “more pure” I do not accept as well. I would say the Christian tradition has only been enriched over the years, and without a unified basic set of dogmas it would really make much sense.