The proof is in the money trail. If the library’s funding traces to a tax-funded government, it is a public service that encompasses all services offered by that institution. It’s also in state or national law that legislates for libraries to exist, which differs from one state to another.
If you want to find a clause that says “only people with wifi hardware may access the internet, and only if they have a mobile phone”, I suspect you’ll have a hard time finding that. At best, I could imagine you might find a sloppily written law that says “libraries shall offer wifi” without specifying the exclusion of others. But if you could hypothetically find that, it would merely be an indication of a national or state law that contradicts that country’s signature on the UDHR. So it’s really a pointless exercise.