Their terms require a phone so yes, on their terms.

I keep a copy of everything I sign. The ToS I signed do not require a mobile phone. It’s an ad hoc implementation that was certainly not thought out to the extent of mirroring the demand for a mobile phone number into the agreement.

Why would they make an exception for anyone?

Because their charter is not: “to provide internet service exclusively for residents who have mobile phones”.

And why would they want to deal with paper agreements for WiFi?

Paper agreements:

• do not discriminate (you cannot be a party to an agreement that you cannot reach)
• are more likely to actually be read (almost no one reads a tickbox agreement)
• inherently (or at least easily) give the non-drafting party a copy of the agreement for their records. A large volume of text on a tiny screen is unlikely to even be opened and even less likely to save it. Not having a personal copy reduces the chance of adherence to the terms.
• provide a higher standard of evidence whenever the agreement is litigated over

You don’t have to be a member to use WiFi, someone else could have given you the password if there even is one

That’s not how it works. The captive portal demands a phone number. After supplying it, an SMS verification code is sent. It’s bizarre that you would suggest asking a stranger in a library for their login info. In the case at hand, someone would have to share their mobile number, and then worry that something naughty would be done under their phone number, and possibly also put that other person at risk for helping someone circumvent the authentication (which also could be easily detected when the same phone number is used for two parallel sessions).

If someone is doing something illegal it’s gonna involve the library if you get caught (that’s why the phone number but maybe they are just being shitty with it). Not worth the risk.

Exactly what makes it awkward to ask someone else to use their phone.