This might be a shot in the dark, but when they want a phone # to send you a text message, will they accept (had you tried) inputting an email address to see if that would work? Technically it can, if they have the open mind to conceive of doing it.
However I don't recall it being the case that the only way to recover a gmail account is a text message. Seems like I have a backup email address set as the recovery for mine... maybe? It sure seems like I did that once upon a time ago, maybe that is no longer an option and I'm grandfathered in? Granted, I have a cell and google knows about it, so maybe they just didn't bother me when they changed that.
I'm the last person to push unnecessary tech on people but no cell phone, really? Even if you want to cut costs, a 2nd hand $20 phone is not that great an expense, and could come in handy now that payphones are virtually nonexistent. If nothing else you can watch videos on it when you get bored waiting somewhere... No streaming required, local storage can hold a few hours of video on just about any phone at the resolution it has... relatively lower bitrate, smaller file size for lower tier phones with lower storage capacity. I'd consider that a hardship but just sayin', a mobile computer that fits in your pocket is not a bad thing.