In Wasabi every participant pays part of the fees. Also, there may only be one coinjoin transaction: depending on your privacy needs, a single transaction can be sufficient. 

Knowing who paid the fee in the first round doesn't give away who pays in the next round.

It doesn't. Wasabi is not vulnerable to that issue at all as every participant participates equally. Nor is payjoin, as it's a payment, combining (some) funds for the two parties.

I'd suggest that you sit down with a pen and paper and try working out what input paid the fees yourself. That's the best way to understand the issue.

privacy

How Good Are The Widely Used Coinjoin Implementations? - Peter Todd

Scoresby

It doesn't. Wasabi is not vulnerable to that issue at all as every participant participates equally. Nor is payjoin, as it's a payment, combining (some) funds for the two parties.

I'd suggest that you sit down with a pen and paper and try working out what input paid the fees yourself. That's the best way to understand the issue.