Numerous people eligible for medicaid don't sign up because they try to and get turned away with some bullshit excuse about not filing the paperwork properly or whatever, either due to innocent bureaucratic bullshit or because the state is explicitly hostile towards government assistance.

With single payer, people wouldn't have to sign up; they would be signed up and have to opt out of they so chose. That's how it should work and it's not hand-wavy in the least.