Update: workaround that worked for me
I claimed this week’s free game in the Epic Games Store from the launcher (Knightica). I didn’t even need to install it. Right after that, the “+” next to Engine Versions started working: it added a 5.8.0 slot with an Install button, and the version list now shows everything from 4.24 to 5.9.0 Preview.
Thanks to the people who suggested this in Launcher 20.3.7: Install Engine and + do nothing, Engine Versions empty, "found 0 entitled apps" (new account). Epic is tracking the underlying bug as UE-413749.