At its earliest, Christianity was a type of Judaism, and one of the earliest internally Christian disagreements, as far as the surviving evidence suggests, was over 1.) whether Gentiles could take part in Christ's salvation, and 2.) whether they needed to convert to Judaism first. The texts that were eventually canonized into the New Testament are all or almost all Jewish texts that, among other things, record this struggle.

A couple of things: First, it's important to remember that in this period, "Judaism" is essentially Judeanness. It's people from Judea or descended from Judeans practicing Judean culture. "Conversion," which had and continues to have certain codified requirements, was literally an act of non-Jews changing cultures, and doesn't seem to have been very common. Imagine a born American going to live in an immigrant neighborhood and saying, "I'm converting to Sudanese." That's a gross oversimplification, but it helps contextualize things. Second, there's a lot of scholarly disagreement about whether and to what extent Judaism was actively proselytized in the first and second centuries BCE. It isn't now for the most part, and even in that period was never uniformly practiced that way, but there are suggestions that some Jews separate from the Jesus movement actively went around attempting to convert Gentiles. There were certainly non-Jews who attached themselves in various ways to synagogues in cities around the Roman empire, but they didn't always fully convert. The usual word we use for those people is "God fearers."

The Jesus movement was an eschatological movement, which means it focused on the end of history. What that means is different in different Jewish contexts--sometimes it just means the restoration of independent and permanent sovereignty, sometimes it's more in line with what we now think of as "the end of the world." Either way, along with that comes the question of what happens to all the rest of the nations when God restores an eternal kingdom to Jerusalem. In more earthly/political ideologies, the nations mostly figure as enemies whose opposition triggers the final war that liberates Israel, and it's not really relevant what they do afterwards. In more apocalyptic strains, sometimes the nations are utterly destroyed, but sometimes they turn to the worship of God, and God's sovereignty becomes centered in Jerusalem but overtakes the whole world. Those are the two ideas we see opposed in early Christian documents--if the nations are destroyed, then anyone who wants to be saved has to become a member of Israel through conversion. If it's possible for the nations to come under God's reign as they are, then conversion isn't necessary. The disagreements you see happening in, for example, Galatians 2, and Paul's whole discourse on the Hebrew Law in Romans, are firsthand accounts of this conflict, where Paul is arguing for the second option.

So why did the second option win out over the first? By the middle of the second century (and I would argue probably earlier), the vast majority of people worshipping Christ were Gentiles. The movement spread quickly, and most non-Jews didn't have any interest in fully abandoning their own cultures, if they even belonged to a demographic that would have had the freedom to do so. There were also a growing number of leaders who were not Jewish and therefore not qualified to oversee true conversions. On top of that, there were conflicts with Jews who did not follow the Jesus movement, and in some places Jesus followers were actively and sometimes violently expelled from synagogues. Basically, you have a massive, multidirectional tug-of-war of identities, regarding who can worship Christ and how they should do it. By the late first and early second century we have texts like Colossians explicitly forbidding Jewish practice (e.g., Colossians 2:16), which indicates that the debate is still raging.

Nonetheless, Jesus worship still then and today largely depends on a Judean understanding of history, where history is conceived as a continuous narrative of God's activity in the world with a specific interest in Israel as his chosen people. Without that you lose the eschatological element, and Jesus is, at best, the son of a god. In the Roman empire that's perfectly fine, if suspiciously novel in how recent and non-elite he was, but also not particularly unique. That, combined with the fact that any tradition without sufficient antiquity was met with extreme suspicion, was the driving factor in the Christian appropriation of Jewish scripture. So the idea that Jesus is the fulfillment of the Law, which Paul puts forth in Romans, became the bridge between appropriating Jewish history and scripture but NOT most Jewish practices. You still get your unbroken line of divine activity and intention, but now (so the argument goes) God has extended his promise out to the entire world by means of Christ, so the specific requirements of the covenants are no longer needed.

To part of your question, by the way, adopting Jewish practices en masse probably would not have helped Christians gain legitimacy. Non-Jews tolerated Judaism to varying levels because of its antiquity, but especially in the diaspora Jews were still very much what we would call a suspect class. Conversion was rare not just because Jews weren't actively seeking converts. Christians, generally, were more hated because they lacked ancestral legitimacy. They had simply abandoned their cultures. They argued that Jewish antiquity applied to them, but for the most part it doesn't seem that Romans accepted that argument, and they could still treat Jews pretty badly, anyway, so it's not clear to me that claiming full conversion would have helped, especially as Christian numbers grew.

There's a lot more to say, and the "parting of the ways" between Judaism and Christianity is still pretty seriously debated, but the TL;DR is that non-Jewish people wanted to worship Jesus without becoming Jewish, but they needed Jewish history to make sense of Jesus.

Paula Fredriksen is excellent for her thoughts on the nuances of this, particularly her book Paul: The Pagans' Apostle. Judith Lieu's book The Parting of the Ways? is also very good. If you want to see the Christian reinterpretation of Jewish scripture in action, Justin Martyr is an excellent primary source.