For the technical solution, I would focus on getting that old phone some power. Wireless charging, gently pushing the charging connector in the magic direction, swapping the battery, taking it to a repair shop that could reflow the power connector or use some kind of battery emulator.
Google just paternalistically chooses which device they think should be your most trusted one, and then insists that you produce it. No consent, no concern for your actual security model, just pure snake oil that tech-rubes eat up as "2fA sEcUrItY". Any damn time I want to use a Google account from a desktop browser, I get the same rigamarole of having to unearth some discharged tablet and wait for it to charge, just to jump through their nonsense. I shudder to think what would happen if I took the obvious remedy and just removed those devices from my account.
I've needed to scrape the USB connector in the phone with a needle (digging around the outside and scraping the tab in the middle) in order to get the cord to stay in place. It gets filled up with pocket lint gunk, and needs cleaning every once in a while. Also since the phone is water proof I've taken to letting water run into the USB connector to really clean it.
