This is a half-truth. Achavi is really a web interface for the Overpass API. But it hasn’t settings for selecting the API server.
However, you can easily fork the repository on GitHub change one URL, and enable GitHub Pages in your repository settings. Or download the repository to your computer and open index.html in your browser.
For an experiment, I changed the server to overpass.private.coffee in my fork
achavi - Augmented OSM Change Viewer [attic] and it seems to work. (There are other references to the Overpass API domain in the code, but I don’t know what they are used for)
p.s. I don’t promise that my fork will continue to work, I’m just demonstrating the possibilities.
p.p.s. overpass.private.coffee is probably not a very reliable server. You can try to specify https://maps.mail.ru/osm/tools/overpass/api/interpreter, but don’t forget to disable Tracking Protection in Firefox