I don't think think legally there are libraries not like that anywhere. To be removed, someone has to be committing a crime. In most cases with homeless people that aren't actually committing an obvious crime it's loitering (or solicitation if panhandling). In a library if you're actually reading a book or using a computer you're not loitering any more than anyone else there.

This is a problem with the design of the system. "Using the computer" just so you can be inside in the air conditioning is definitely not how those resources were intended to be used. The problem is that it's impossible to practically police this behavior individually, but easy to detect in aggregate. If the homeless person just sits around in the library doing nothing, they're loitering, but because they're sitting at a computer eating up a resource, it's not loitering. I certainly don't mean to assume a homeless person wouldn't have a legit reason to use the library computers. That's why I noted the difference between policing an individual is impossible, but seeing the trend in aggregate is easy.

But we require our students to talk a lot. At least in the libraries I've been to there's an expectation of noise level. That said, I haven't been to very many libraries.