The Messianic Jews say the Old Testament food laws were never repealed. I also heard on the Bema podcast that part of Mark 7:19 ("Thus Jesus declared all foods clean") shouldn't be in the Bible—that it was added to the original text.

This is simply not true. The words in question appear in the best ancient manuscripts. Let's examine the text.

Jesus says, "There is nothing outside a person that by going into him can defile him, but the things that come out of a person are what defile him.” ... And he said to them, "... Do you not see that whatever goes into a person from outside cannot defile him, since it enters not his heart but his stomach, and is expelled?” Thus he declared all foods clean (Mark 7:15, 18-19). This was not necessarily important before Pentecost, but for the early church, which was making a break with Judaism, this was significant. Whether Jesus was abolishing kashrut at the moment he spoke these words, or only in principle—with validity only under the new covenant, soon to be set in motion—is less easy to determine, though I would opt for the latter interpretation.

As commonly understood, Jesus seems to be invalidating the kosher rules. However, this interpretation is far from certain. It is more likely, esp. given the context as he addresses a Jewish audience, that he is declaring all (kosher) foods clean, whether or not hands are washed. For more, see Joshua Strahan's article "Did Jesus Nullify the Torah and Declare Nonkosher Foods Clean?" in The Bulletin for Biblical Research, vol. 33, no 3, 2023 (259-280).

Further, passages like 1 Tim 4:3-4, Heb 13:9 and Col 2:16,21 do not support the kosher interpretation. Certainly we benefit from knowing the Jewish background of the New Testament, the apostles, and Jesus himself. However, kosher laws are no longer part of the covenant. (For more, please refer to the series on Messianic Judaism.)