Saying how much you want is different from saying how much you currently make. It may be numerically the same, but there are big differences in how that information is treated. One of several examples: A recruiter who has been told to offer you up to 100k will probably not go back to the hiring manager if you just say "I want 120k"; they will tell you that's outside of the range for that position. But if you say "I make 120k and would need to earn at least that to consider leaving", that is much easier for them to escalate. This is something I've seen happen at two different companies, on the hiree and hiring manager side.

The background check thing feels mostly like a boogie man to me. It's convenient for recruiters that engineers believe that, but I've never seen it happen in practice. And if it did (how does this even manifest? they require you to send them a paystub? they can't compel your previous employer to divulge this info), you should be outraged! It's none of their business what your paycheck at your previous employer looked like or contained. I would respectfully decline to provide this information, and if they insisted I would edit or photoshop it, and give serious thought about whether this is the type of employer you want to work for. You should be indignant about the fact that an entity you are planning to devote the majority of your waking life to for the next few years is comfortable with systematically fucking you over like this.