Online exchanges like ChangeNOW, SimpleSwap and FixedFloat, unless they keep records for a long period, which theoretically might be later accessed by authorities, might offer considerable privacy. I used one of these, with the Tor Browser, to exchange a coin I had bought on a KYC exchange to another kind of coin that then went to my personal wallet. No information other than the two addresses was provided to the exchange. I took a chance by not providing an email address that I might have difficulties if something went wrong, e.g. losing my internet connection, but the exchange also provided a case number and, in fact, I had no problems.
Am I missing something, or does this provide anonymity to the bitcoin that ended up in my wallet? This might be something that one would not like to do with a large amount of coin, but that was not a problem for me!