If ChipMixer keeps logs (which you can't possibly know for sure), then they know exactly where the coins are coming from and going.
Does there any way to check if it really does?
This is not possible. You must trust the operator. If the operators servers are compromised, a third party may create logs without the operators knowledge.
Binance should also work fine, right? Even though they require registration, I think you could easily use fake information and just create multiple free emails through email services like ProtonMail[1]. As far as I know they don't lock suspicious accounts, though I'm really not sure.
[1]
https://protonmail.com/Dont try this. I can assure you binance looks at much more than your email address. Doing what you describe is a good way to be required to provide a lot of KYC information that will probably be passed along to your government in the form of a SAR report, in order to access your coins.