Wood doesn't conduct, so you're right that the connection is doing nothing. A ground wire's whole job is to provide a low-resistance path back to the panel so that if a hot wire touches metal somewhere, the fault current has somewhere to go and the breaker trips fast instead of the metal staying energized. A screw biting into a pine stud gives you basially zero continuity anywhere. Dry wood is an insulator, and even damp framing lumber isn't going to carry fault current in any useful way. My guess is what happened is somebody was splicing or extending the circuit, needed somewhere to stash the ground, and just parked it on the nearest thing with threads. I've pulled apart a few older houses and found similar nonsense, usally from a handyman or a previous owner doing thier own wiring. No electrician I've ever worked with would do that, it's not an old acceptable trick, it's just wrong.
The fix depends on where that wire actualy runs. If it feeds a box or a receptacle somewhere, the ground should land on the ground screw in a metal box, or on the ground terminal of the device, and continue back to the grounding bar in the panel. If the wire is a dead end with no home, the cleanest move is to trace it back and see if you can just cap it off or remove the run entirely, assuming nothing downstream depends on it. As for your tripping problem, I wouldn't assume this is the cause. A floating ground usually won't trip a GFCI on its own, since GFCIs compare hot and neutral current and don't actually care about the ground for tripping. More likely you've got moisture somewhere, a nicked conductor, or a device on the circuit leaking current. Fix the ground situation anyway because it matters for shock protection, then start unplugging things on that circuit one at a time to narrow down the tripping culprit.