Five years ago three of us flew into Xining and onward to Golmud by train. Absolutely no issues with visas at any point, ie we had 30-day normal Chinese visas.

In Golmud we booked bus transportation, visa extensions and hostel in Lhasa. Once there we booked transport to Everest Base Camp (Tibet) and on to the Nepal border.

Coming up through China to Golmud over about a week allowed us to acclimatize and none of us had altitude issues.

Things may have changed - but I'd bet that Golmud still has TAs specializing in transport into and visas for Tibet. And I suspect getting train seats back from Lhasa is fairly simple.

Happy wandering

Fred