Christianity understands itself to be heir to God's promises to Israel. From a Christian theological standpoint, it's not that the Old Testament is "Jewish" and the New Testament is "Christian" and those are two fundamentally different things, it's that both testaments represent a continuous line of God's activity in the world, which reached its climax with the death and resurrection of Christ, and will be fulfilled at the Second Coming. From that standpoint (which I am explaining, not endorsing), what we now call Jews have rejected God's plan for the universe, and therefore are no longer the true Israel.

Early Christian literature is full of references to Hebrew scripture, especially prophecy, in an attempt to establish this link. The book of Isaiah is a major favorite. The "suffering servant" passage (Isa 52:13-53:12), which was originally in reference to the Babylonian captivity and probably refers to the people of Israel as a corporate entity, shows up in Luke, John, Acts, Romans, etc., reinterpreted to prefigure the suffering of Christ. Justin Martyr, in the mid-second century, goes on for PAGES quoting giant chunks of Isaiah and interpreting them to show that they are talking about Christ and Christianity. Paul makes a big deal about Abraham in Galatians 3 to try and demonstrate that the promises to Abraham in Genesis were, in fact, about Jesus (which leads to one my favorite Pauline logic pretzels in Gal 3:16).

Part of this is that there were ethnic and pious Jews (like Paul, Peter, etc.) who genuinely understood Jesus to be the Messiah, in a long period when there was a lot of political turmoil and social change, and therefore competition for what it meant to be a "good" or "real" Jew. Stuff in the New Testament that we read as horrifyingly anti-Semitic is often pretty comparable to the way that, for example, the author of 1 Maccabees (a Jewish text 200 years or so older than the Gospels) writes about other Jews that he disagrees with (e.g., 1 Macc 1:11-14, but it's not limited to that passage). The Essene community was also incredibly unflattering toward the Jerusalem priesthood. So Jesus worshippers weren't the only ones claiming to be the "real" Israel among a bunch of apostates. The difference is that their movement very quickly spread into the non-Jewish world, and so what had been pretty standard internal identity competition in a politically volatile period (think about the ways we talk about what a "real" American is/should be, or whatever your equivalent if you're not American) very quickly became an external power move. Nowadays it would be a genuine example of cultural appropriation.

TL;DR: Christianity doesn't tolerate the Old Testament even though it's favorable toward Jews. A better way to frame it is: Christianity appropriated Hebrew scripture and claims to be the real heir to its promises, excluding (non-Christian) ethnic Jews from consideration.