Yes they do.

Should they be

That's a loaded question. Clearly my local community thinks they should.

I'm pretty sure my library does. They also maintain a "Library of Things" where you can check out a 3DS or a Playstation 3 for the weekend. None of the newest stuff, but a decent selection nonetheles.

Might be a funding or Board of Directors issue with your particular library?

My local library has PS4 games and a room that has the console. You can use the room, or loan out the games like a book, except it's for like a week instead of the whole month. Of course there are no PEGI-18 games and other age limitations are followed too. There are often kids in the library, in that room or in the other areas. Not many seem to be into the gaming, more into playing, giggling or reading kids magazines and such, at least what I have seen and heard.

I think it's alright, games are culture too. But if someone is not into games, I don't think they're any less "civilized."

I think it’s totally reasonable. I think there should be a whole games library separate from the books libraries so it doesn’t get too loud. I never went to church but from the outside as a kid it acts like a gated community rec center, with basketball courts and playgrounds, and all that god nonsense. I’m assuming there were tabletop games inside most too. I think it would benefit everyone to have access to something like that. Book libraries are like solitary third places, but a games library could be a more sociable version.

At least in the UK they have done for decades. No idea about where you are.

Where are you? A lot of the Milwaukee libraries here have rows and rows of console games (though, sure, never any PC).

Yes but the libraries lend them and the people borrow them. Rent has a nasty air of money exchanging hands.