[swarmee]

OK, here are my thoughts.
1) If you are worried about a Pete Rose rookie in a PSA slab, send it back to them on a "reholder" submission. They will confirm the card is authentic and was the one put in the slab (that it was not cracked) and then move the card to a newer, more secure slab with the newest flip (hologram and cert numbers both sides).
1a) You could also have the card sent in for "review" but it may come back as a lower grade.
2) You can do many searches on this site and others about why PSA is currently the highest regarded TPG PRICE-WISE. It mainly has to do with the following: market share, preponderance of collectors of 1950-1990 cards using them, and the PSA Set Registry, where collectors can display their collections and compete with other collectors for having the "Finest" sets of all PSA graded cards.
3) I have only collected graded cards for two years, and I've never seen someone else claim BGS/BVG to be the most stringent grading company at any time. But I currently own 500 graded cards, the vast majority being self-subbed PSA cards.
4) Sending BGS cards in slab to PSA will likely not get you a good percentage of cards to cross over at the same grade. You will likely have to break them out of the slabs and send them raw, but it's possible your cards will lose grades or not be graded due to trimming, or not meeting the correct size for their issue. Take a few months before deciding to send all your cards to PSA for crossover.
5) Welcome to the Net54 boards. We can see you're excited.