In theory it should work, as Stable should have the new connector. Maybe because they adapted to Manifest V3 they had to drop support for ESR. Though they would have written that in the changelog. But you can easily test that, Mozilla has their own Debian repo now. Install the normal Firefox from there and try with that.
As a workaround, you can install the Extension Manager as Flatpak, this will allow you to install and manage extensions - compared to the preinstalled one only being able to manage them.
Best