I don't really know tárogatós much, but I'm guessing this is similar to clarinet metal socket rings? If that's true, I prefer to tighten them by gluing them with shellac. Here is why.
I've glued more than a few rings with shellac, in places where temperature can vary from around 0 to 40 (C) and humidity varies from my very dry city to the very humid city I often go to with my clarinets. I've never had problems. I have one on my own clarinet that was very presistantly loose in the winter (relatively tight in the summer) and it hasn't moved for more than three years. It is a good and stable method. Same for others I repaired using this method.
Dried shellac is hard and offers very good support for the ring. You could also use a type of other heat melting glue, but those are slightly softer and more flexible, so offer less support. Shellacs can vary in how brittle they are, so a very brittle type is not as good. Some claim the shellac is too hard and solid so it will crush when the ring tightens, but after a lot of experience with this method, this doesn't happen. For example I never have any issue with my own shellaced ring.
So in short, IME shellac doesn't have the issues that are sometimes suggested it would have.
As far as advantages over other methods, IMO there are several.
With shellac, the exact amount of "shim" is put between the ring and the body. Both the ring and body are probably not 100% exactly round. Putting a "fixed" thickness of shim like paper, fabric, etc. might not be the best all around the ring. It might make some areas tight but leave other areas less supported. Shellac is like an automatic perfect thickness shim. In reality, there is noreal way to know how much this matters, but there is no disadvantage from using the best thickness shim. I have seen the other methods for shimming slightly distort sockets and they might not support the body in a way that would necessarily help preveting cracks.
As far as physically reduce the size of the ring (there is a tool that does that), IMO it has the same disadvantage as above. It is also harder to remove and change again if needed, for the example if the player moved to a different area. With the shellac method, it's easy to remove the ring and re-tighten it again in a different area if needed.
Similarly to the other method, it is a fast repair that can be done on the spot and it doesn't require any expensive or special tools (unlike the ring press) so it's an inexpensive repair too. Basically, IME no disadvantages at all, only advantages.