Did you take the time to read the privacy policy of the mixer you are using?
You're correct, and yes, I have taken the time to read their privacy policy, which is why I'm not endorsing anyone to using this particular mixer.
Your mixer says it can confiscate your coins or terminate your account, so the only difference between using an exchange and using your mixer is that exchanges have a larger user base. It is not suitable for the purpose of gaining privacy.
There is another difference: the mixer acts as the liable entity for tumbling everyone's coins together. You don't gain this from an exchange.
This is why you should absolutely not be recommending mixers to the OP as an alternative for exchanges.
The previous mixers advertised in this place did not have the same privacy policy and never blacklisted anyone.