It does not eliminate code reviews.
In practice, you should have at least one independent reviewer who did not actively worked on the PR.
That reviewer should also download the entire code, run it, make tests fail and so on.
In my experience, it's also good that this is not a fixed role "the reviewer", and a responsability everyone in the team shares (your next task should always be: review someone else's work, only pick a new thing to do if there is nothing to review).
This practice increases quality dramatically.
> It does not eliminate code reviews.
Yes it does. There are many ways to do things, of course, and you can institute that there must be an independent reviewer, but I see this is a colossal waste of time and takes away one of the many benefits of pairing. Switch pairs frequently, and by frequently I really mean "daily," and there is no need for review. This also covers "no fixed responsibilities" you mentioned (which I absolutely agree with).
Again, there are no rules for how things must be done, but this is my experience of three straight years working this way and it was highly effective.