Totally not a dumb question tbh - the answer probably also highly depends on where you’re from.

In Germany (also highly depends on county/city) yes, they do normally fare pretty well and shelters often operate multiple feeding stations where there are a lot of strays. So they get to know the „local strays“ and can keep an eye on them via volunteers and cams.

When a cat is visibly sick, they catch them at this station, treat them and if the cat is alright, oftentimes release them again - shelters are notoriously full and money ist scarce for these organisations unfortunately…

However, if there are conditions that make life for this cat (or others - e.g. FIV) dangerous, they are normally kept in the shelter and then opened for adoption as a „only indoors cat“.

Also, older cats often end up in shelters because they often belonged to older people - and when they can’t care for the cats anymore, have to move or die one day, the cats have to go somewhere too.

Where I am, strays are usually catch, sterilize, release. But if you caught a cat who was sick they'd treat it, and lots of cats end up at the shelter because people don't fix their pet cat in time, and they get pregnant (even if you are responsible - we had a brother sister pair and I asked the vet what she would he fix them, he said "6 months", I asked but when can they get pregnant? "4 months". Thankfully it didn't happen, they were fixed and lived long kitten free lives).

And pets, if someone gets sick and can't care for them they get surrendered.