Read through the links, and some of the links shared from those messages (really need to get back to work now).
For posterity, I'll provide a summary:
CONS
I read a lot of concerns about rentals, summarized as
1. People drive rentals like, well, cars they don't need to care about
2. The rental company will delay maintenance to keep the car out making money
3. The rental company is slow to get recalls done
4. The CarFax is unreliable as they do their own maintenance and don't report most issues
These people have not bought a rental, though the concerns are valid in my opinion.
PROS
1. If you're a used car buyer, you're taking a risk, period. At least there's a maintenance department responsible for these vehicles.
2. The best rentals are sold directly by the rental agency.
3. Everyone who replied to those threads that had real experience with buying a rental direct from the rental company said it was a good vehicle, and many added something like "would do it again".
I am beginning to think that I will ask a rental agency if I can rent a 12-passenger van they are selling, and apply the rental cost to the purchase if I decide to buy, During that time I'll get to test it and bring it to my own mechanic for an inspection.
If I go down this route I'll be sure to circle back here, hopefully making this a useful post to future readers.