Finally fought my fear, snapshotted the config directory, and updated to hacs 2.0. Everything seems to work except for the “Update Information” in the three-dot menu. It does not work in neither of browsers, I cleared cache, relogged-in - nothing help. Surprisingly, reinstall menu works just fine.
Oh wait, so this is not a replacement of the old “update” in hacs 1.0. Okay, updates are now in settings->updates. I never ever used this menu because I manage versions of my software external to HA. My bad lol.