Seems to me that they’re are easy solutions to your scenarios.

Ask to use their WiFi that isn’t overloaded. Get a working SIM before you travel (eSim perhaps?). Download the one time code and screenshot it. If it’s a code like the DVLA driver license check then it doesn’t matter how you’ve saved it. Only that the code is valid when they enter it. Write it down on a piece of paper if you have to.

Ultimately it’s on you to understand the requirements to travel to any country you visit and have the correct paperwork. And if you can’t do that… well 🤦‍♂️.

If border control pulls me aside and demands my proof of residency, and I can’t access the internet, I may be stuck in a small room for hours or days,

How likely are you to not be able to access the internet at a UK border you might be traveling through? Seriously.

Also, have you heard of a download for offline usage?