Actually, it appears the very earliest Christians did. The Book of Acts shows Peter presenting Christian teaching to Jewish crowds not a new religion but as additional acts of the same God they followed:

The God of our fathers raised Jesus, whom you killed by hanging him on a tree. God exalted him at his right hand as Leader and Savior, to give repentance to Israel and forgiveness of sins. And we are witnesses to these things.

Of course, many Jews vigorously opposed this, both in Jerusalem and throughout the Diaspora. The name "Christianity" was given soon afterwards, though it doesn't seem to have been immediately adopted; it only appears three times in the New Testament itself, and could be used simply as a name for a sect of Judaism rather than a separate religion. Years later, Paul - after having preached Christianity among Gentiles - is still reported as saying, "according to the Way, which they call a sect, I worship the God of our fathers."

Christianity and Judaism parted course shortly afterwards; it's still debated exactly when and how. (Here are two previous threads exploring the subject.) What's clear is that the parting was not immediate; there was a period of time when Christians indeed did consider themselves, in your words, "true followers of Judaism."