They'd lose a lot of business if they did that with any random Joe
Except that they already do, and lose minimum business? They don't make it seem an extraordinary claim, but that's what Binance, Coinbase, Kraken, Gemini etc., all essentially do. If we take the crazy assumption that taint analysis isn't pointless and violative, then the term that you'll have your funds confiscated if the indefinite algorithm says so doesn't sound pointless and violative either.
pretending without bothering to do an analysis at all would make them a scam
Let's take Gemini as an example. From their user agreement page:
As such, we have a Gambling Policy that prohibits the use of Gemini for storing, sending, or receiving wagers or proceeds of Illegal Gambling. Proceeds derived from trading contracts for differences and spread bets on licensed and regulated platforms do not constitute Illegal Gambling or a violation of our Gambling Policy. We employ a number of methods to detect violations of our Gambling Policy.
You agree and understand that we have the right to immediately (i) suspend your account and any related account, (ii) freeze/lock the funds and assets in all such accounts, and (iii) suspend your access to Gemini, if we suspect, in our sole discretion, any such accounts to be in violation of our Gambling Policy, which is a direct violation of this User Agreement.
From Pmalek's "What Do Centralized Exchanges Consider as Taint?" thread, this is what Gemini responded:
Translation: We, Gemini, have every right to shut down your account, confiscate your property if we suspect your coins come from illegal gambling activity. We have a number of methods to detect such activity, but we're not going to reveal them. Your coins might come from such activity, and you might not even know it. In any case, your account will be closed.