In that case all Christians should be by faith be Jewish(?). And if the above is the case then in Jesus' time the cross, the blood of the lamb and the usual Christian message we hear today, and all that, didn't apply so again all Christians should be Jewish, and that Christianity is a false religion built on idolatry.
OK, let's take a more up-to-date analogy. In the 17th, 18th and 19th centuries, Britain basically had two political parties, the Whigs and the Tories. In the late 19th Century, parts of the Whig Party changed a certain amount, but retained their fundamental ideas, and became the Liberal Party which sought to fulfill those fundamental ideas in what they deemed to be a better way.
This is an analogy of Judaism and Christianity in so far as Jesus clearly came to the Jews initially and sought to move them out of what had become a very introverted and insular faith, under which it as thought that Jehovah was purely a deity for the Jews and no-one else - which contradicted what they had been told that they had been chosen to do. Sadly, the majority of the Jews of the time either rejected his claim to be the promised messiah (predominantly the leaders of the faith) or simply didn't understand enough to make decision.
Even during his lifetime, Jesus had begun to take the message of the Gospel beyond the 'boundaries' of Judaism, and he effectively reinstituted the original instructions to the Jews (recorded in the Hebrew Scriptures) to take the light to those beyond their physical and tribal boundaries by instructing the disciples (not merely the 11 apostles) to 'go into all the world and make disciples'.
Instead of following the message they followed the messenger.
No, they followed Christ's reinstituted and, yes, developed instructions, but continued to carry out the original instrctions to worship God.
Why was it a term of abuse?
The early Christians were, like the Jews, monotheistic which was a very unusual thing aroujnd the Mediterranean of the time, and the people of Antioch were alert enough to realise this but that they based their lives on the teachings of both the Hebrew Scriptures
and the teachings of the character they called 'the Christ'. As with most modern-day societies, people who are different get given nicknames, many of which aren't meant to be complimentary.