I went to 4 shelters when I was looking to adopt my cats. The first 2 were very small and private (about 10-15 cats), the 3rd quite a bit larger (about 35 cats) but still private. The cats weren't confined to cages at any of them. At that larger one, they were in a very large room with multiple litter boxes and feeding stations and most were lounging around. On the one hand, it was great not to see them in cages. On the other, it was a turn-off to me as a potential adopter. It smelled like cat urine and just felt dirty. I'm sure they kept everything clean, but with that many cats, you'd have to scoop or change the litter and vacuum pretty much constantly. Also, it was hard to see all of the cats. The timid ones would hide, and some would move as you moved around, so you couldn't necessarily keep track of which cats you'd already seen. It was like cat overload, and I ended up spending not more than 5 or 10 minutes there. That sort of set-up seems to me more suitable to a sanctuary than a shelter that wants to adopt out animals.
I did adopt my first cat from one of the small shelters, in a very rural area. Honestly, I don't even know if they had the cats in cages or not, though the large dogs were in a large pen outside. When I said I liked tabbies, they took me to a separate little building with about 4 kittens inside, roaming free. I don't remember why they were separated like that.
I believe that at all 3, they took care of s/n before the animals were adopted out.
The current 2 were from the local SPCA, where the cats were in a huge room, all caged. There were easily 70-80, and one had been there for 2 years.
But they also had a cat room, where the kitties who got along with the other kitties were allowed to play for a certain amount of time each day. The adoption fee included the price of s/n and microchipping. Once you paid, they scheduled you for the s/n.
My last vet had a low-cost s/n program - I think it was once a month. I don't know about the current vet, but I'm going to ask.