i have managed to get most letters as email/SMS links to the (annoying/unreliable needs some UI work) Dr Doctor system for most correspondence. This is a huge access win for me. However I do still get some postal letters (I think it's secretaries vs central appointments) which means the e-system isn't complete... Need to chase GP about full historical content access in NHS App.
An option for a snailmail letter is not to use nasty plastic windowed envelopes (harder to recycle) and have a fully paper envelope with address on it separately. This is how hospitals around here do it.
My ortho hospital asks at clinic checkin on the stupid screens (they have other issues and bad touchscreens and terrible UI) if you are OK having your name up on the screen or not alongside the racist immigration and 'we don't actually comply with the AIS' sensory impairment questions.
The name calling screen doesn't say which doctor Named-Person is seeing just "go to rooms xx-to-yy" down a large corridor which is the clinic-area. The rooms are not consistent appt to appt, although I can guarantee as a mobility impaired person that when I have the most pain it's the furthest one (sodding Murphy)!
There is a separate screen with clinic names/doctors and approx waiting times.
So there's clearly hospitals who have thought of many of the issues you raise. None which would arise at this hospital!