could be a simple “redirect to a different instance” button, where you could select yours

Though not perfect, that would be a vast improvement on now. There are thousands of instances (remembering content is shared to Mastodon and others), and not every instance knows about every other, but you could auto-fill from the list the instance knows about, and if you end up back on that site it should remember what you selected last time and prefill it. Great idea!

Or the site could check for the cookies of known servers

This is simply not possible because browsers don’t let a site do this.