>UK is in this weird place where there isn't one kind of ID that everyone has - for most people it's the driving licence, but obviously that's not good enough

The US also lacks a national ID, but as a non-driver myself, this is handled by things called variously by state a "state ID" or a "non-driver's driving license". These look exactly like driver's licenses and can be used wherever those can for ID (like for flying) except for a line saying "not valid for driving".