The answer is as simple as "because it isn't designed to do it". Exchanges share tons of stuff, which user owns which addresses, trading volumes, etc., with governments and chain analysis companies. As I've already shared, there's even a site[1] which takes as input an address and returns you to which exchange it belongs. It's also more expensive.
Despite the fact that exchanges, as you said, collect a lot of different information about the user, in some cases they have some advantages over mixers.
Firstly, if someone is not very afraid for their privacy, using an exchange as a mixer is corny cheaper, easier and faster for an ordinary crypto user.
Secondly, exchanges help clean up bitcoins for AML. Before that, I actively tested the whirlwind mixer, now I'm testing MixTum. I can state with all confidence that after using the mixer, it will not be possible to use bitcoins everywhere. Some services accept them, while others simply do not pass the AML check. In this case, the exchange comes to the rescue, which bleaches bitcoins after the mixer.