With covenants it would be possible to make the round protocol more privacy-preserving so that the server cannot link the inputs with the outputs.

It's already possible to do this without covenants. For example, Wasabi's coinjoin coordinators cannot link the inputs with the outputs.

We came up with a way to do it without covenants as well, but it comes with some extra work required for all users during all rounds, which can add latency to the already fragile interactive process.

I'm already aware of how you could add privacy to Ark right now without using covenants, my question was "How would covenants make adding privacy to Ark easier?"