Is there a good way to bond wood to brass, in strong and permanent way?
In this context, no. There's too much available torque to break a glue bond and once you get over the initial resistance you're in much the same position as if you'd applied no glue*.
It would be very different if you were bonding a flat piece of brass to a wooden surface. Even PVA and other glues not known for their ability to bond metal can work surprisingly well in that way.
There are I think only three reasonable candidates for adhesive here, superglue (cyanoacrylate), foaming polyurethane and epoxy.
Very strong in a thin layer but doesn't resist shear well as you note yourself and even the smallest rotation of the brass ring results in a shear force being applied to the glue line. While it could be strong enough to resist turning by the fingers of a child one sharp knock would be enough to crack the glue and render it nearly valueless here*.
Worth trying if you already own a bottle since you lose nothing if it doesn't work well enough.
If you don't already own any and had to buy it for this job though you'd have to make a point of using it up quickly after breaking the seal as it can, and does, go off in the bottle sitting on the shelf — it's not at all uncommon for the great majority of a bottle of polyurethane to go to waste in this way and the glue is not cheap.
Although polyurethane is known for its strength I don't know how secure the bond would be here and anyway I think there's a better candidate.
Your best bet of these three I'm sure. Even a basic 5-minute epoxy is surprisingly strong but generally the slower the setting time the stronger the epoxy.
Slow-setting epoxies tend to be more expensive but at least epoxy has a long shelf life (nearly indefinite IME as long as it is tightly capped) so the unused portion left in the syringes is sure to still be usable somewhere down the line for any future jobs that might crop up.
*I say nearly because almost any glue will give some benefit in that it will strengthen the short wood fibres present in the threading in the hole, as well as acting to fill tiny gaps. So even if the glue bond is broken together these will make the brass screw eye much harder to rotate which is a definite benefit.