I'm not saying you are wrong by any mens, but I have not experienced this personally.
You may have had, and you don't know. I have come across lots of merchants, one of which was sending me addresses to deposit, and until I asked, they were using Binance. So, I do have sent coins to exchange.
Well. Let's clarify. A man sends over TOR/VPN his coins to exchange which doesn't require him to go through KYC procedure. Which private details could be revealed in this case?
Which part of "it isn't designed to offer privacy" don't you understand? Of course and you can make an account anonymously, but that isn't my point. Your coins could be confiscated for either being "tainted", or your account could be terminated for using a "banned IP address"
Har har, the same could be applicable to mixer that, eventually, bows and scrapes before government's whims and wishes. Anyway, the discussion has started from the point of privacy point (which was the main concern of OP) and slithered to safety. Mixers are not safe at all. Whirpool is case in point. Exchanges and mixers are equivalent in regards of their commitment as to safety of clients money, IMHO.