Well, let's say somebody has 1000 BTC as a single input. Maybe he doesn't want to tell the CEX he has 1000 BTC and a mixer does that even if only 0.1 BTC is mixed and the rest is kept "clean".
This is the distinction I'm talking about. One may have 1000 BTC, another may not want to reveal he's the owner of a hundred UTXOs, another may not want to reveal he's mined a block himself, another may not want to reveal he was a user of another exchange before etc. But, that's a private matter.
The "taint" nonsense is a political matter. CEX's ask for KYC, sell info to basically everyone, mark some coins as "dirty", not because they will prevent money laundering, tax evasion etc., but to gain data, which is ultimately a weapon of control. If everyone abruptly refused to continue using Binance, because of this, be sure that they'd change their ToS before tomorrow morning.
It's a smart decision indeed, and I think that you're one of the lucky few that could actually do that. I think that the majority just won't. Unfortunately.
Better late than never.
Of course, if one sends 1 mixed BTC to CEX in one step, he's "asking for it".
I don't think that's an English idiom, but I get what you mean.

It doesn't have to be a mixed one. Your coins may be "tainted" and you may not have a clue. Just don't use a CEX, end of story.