I wrote one for personal use. If you (by the book) reject invalid feeds you lose more than half. Attempting to properly negotiate the content would just result in failure for each feed (could be many) but you could try it if all else fails I suppose. (without much result but if you try 100 such things you will get to brag about rare successes)

I think (only) if a popular service required it it could be a thing.

Speaking of Atom/RSS:

There's no reason to have a separate /archives resource and a /feed.xml (with or without content negotiation). You can just specify some external XSLT with an xml-stylesheet processing instruction in your feed XML that will cause the feed to be rendered nicely when it's opened in the browser...

Is there any site that will provide an Atom feed (and not an HTML page) when asked with 'application/atom+xml' in the accept header?

I've tried Github release page, but it doesn't work.

For RSS Atom "application/rss+xml" is non-standard

Probably only a few, but there are web servers out there where it would be a minimal effort to implement it, which are sadly only a few too.

It's always confusing me how a Feed Reader looks like a normal desktop browser when it loads a feed, e.g. Accept: text/html instead of application/rss+xml first.

Most feed readers don't even tell the server which format they handle at all, most just send a Accept: text/html, /

This is planned for csvbase at least.