PSA is by far the most reliable company for grading packs at this point, but GAI had a big head start, so most of the product out there is graded by GAI. GAI started out reliable (Mark Murphy was the one who started the company) but went off the rails at some point. Steve Hart of the BBC Exchange is in charge of weeding out fakes for PSA and is both honest and competent.
I don't own a 1971 cello pack, though I will someday. I'm assembling a run of wax, cello and rack packs from 1971-1980. The 1971 is the only cello pack I'm missing, while I still need racks from 1971-1973.
You're right that SGC is preferred by pre-war collectors, though PSA seems much more popular for post-war. I know little about pre-war stuff, though with the REA catalog for bathroom reading I'm learning!