Thanks for the feedback!
Replacing <a> tags with buttons was originally on my list, but I've been pushing it back as there are some additional considerations here:
There are some "proper links" (such as the admin link) that should remain that way. As such, this wouldn't apply to all links automatically.
When the modal ("slide overlay" in module settings) mode is disabled, almost every item should be a regular link – except for browse, which technically shouldn't do anything there, as in that case it's just an indicator of your "current state".
Admin Bar should work regardless of JavaScript. This makes it more robust, and as such more accessible.
That last point isn't an argument against using buttons, just something to keep in mind when making changes: everything should remain functional regardless, which could mean links by default (when JS isn't available, or working as expected) and buttons otherwise. Since default styling is now based on BEM, it should make this easier in some respects, though; technically what element is used shouldn't matter anyway.
I don't actually see (m)any notable issues in the current behaviour. None of the built-in Admin Bar features will create any content without user actually submitting a form – and I'd be very surprised if a plugin did that on itself. The logout link is problematic though, you're absolutely right on that one – I've added this on my todo list, but not entirely sure yet how to handle it.
I'm not familiar with Turbolinks, but the way Barba.js usually works is that you define a container, which you then switch with new content. I'm not particularly experienced in this regard, but to my best understanding you wouldn't usually define your entire <body> element as a container, and since AdminBar attempts to place itself as the last element right before </body> closing tag, normally this shouldn't be an issue – though again, this is just based on my (quite likely lacking) understanding of such libraries ?