Thanks for the reminder about Sabine Hossenfelder's paper. It provides a pretty good overview of the phenomenological (testing) possibilities of various QG.
When the subject is quantum cosmology I am trying to train myself not to automatically think of the world as divided between the small scale (quantum) and the large scale (gen. rel.) but instead to distinguish between high density and low density.
You may think of the universe as "large scale" but who knows? The portion we currently observe was presumably very small scale at the beginning of expansion. But the whole thing could even have been infinite at the start. It seems we don't have much of an idea about the overall scale.
We can estimate is the density at early times. I suppose that it is that which decides where quantum cosmology applies.
So I try to think of QC as the physics of very high density
(It's how one imagines the universe, neither especially large or especially small in linear size, perhaps, but dense.)
Aside from that petty quibble about phrasing, I agree with the spirit of your remark that
" The large scale ones are important too, of course."