The latest flatpak update I did pulled in a new Steam runtime. With this new runtime, Steam starts but the UI is broken. The Store and Library tabs are blank. The downloads window does show current downloads, if any. This makes it impossible to play any games in flatpak Steam.
FWIW, this was fixed by having Steam manually check for updates. After a new Steam runtime was downloaded through the flatpak update, the Steam UI was broken as mentioned in the original post, and every time Steam started it did not ask to install any updates. I ended up checking for updates manually, and an update was downloaded and installed; upon restart, Steam works appropriately.
It looks like Valve has changed the Steam UI again, and I think that this change combined with the flatpak steam binary update led to this broken state, in which it was not even automatically checking for updates as it usually does.