I have some older homes that were always painted and a good oil enamel would last decades.
The problem I run into now is a quick latex roll out over oil or varnish and the top cost just peels off with a finger nail.
Worse is in areas such as a laundry room or steamy bathroom where all the paint alligators… often simpler to go over with new rock.
In our current house, the entire house was wood. No plaster. No drywall. All wood. And not a lick of it matched. At a minimum, there were 3 different woods in a room. One had 7. It was not nice. Think stained rough plywood for walls in the bedrooms.
Our living room had maple floors, 12" white ceiling tile (not asbestos, had them tested), knotty pine shellacked vertical board walls, very dark doors, and very dark built in cabinets on one end of the room. One curio cabinet on the right, and a gun cabinet on the left, with a bench spanning the area between the cabinets. Drawers under the cabinets and bench. Nothing matched. 3 of the four walls had the same style of knotty pine boards (differing width), but the fourth had them all the same size.
I gutted the bedrooms and dining room and drywalled, and put up oak trim on everything, but left the living room alone, because I liked knotty pine. This was supposed to be our 5 year house.......
20 years later, my wife was tire of the living room looking like a mismatched cabin. So was I. So we decided to paint the shellacked knotty pine and make all the trim match the cabinets and doors. About the only real way to do that and not worry about latex paint peeling off is to use TSP to wipe down the walls, removing grease and oils, then sand the shellac to give it a texture, then prime with appropriate primer, then paint.
So I spent about 4 hours a night for a week or so with a palm sander hooked to a shop vac getting every little crack and joint and flat surface roughed up. Primed and painted the color she wanted. OK.
Then I installed new door and window trim and small crown molding. Then came up with a formula for a stain that matched the existing dark wood cabinets and doors (rosewood and walnut, many tests until correct).
So now the room has painted vertical boards and all the trim matches the dark doors and curio cabinets. The maple floor is a nice contrast to the dark trim. The small dark crown molding frames the white ceiling and makes a nice line between the wall and ceiling. Wife is happy. She gets paint. I'm happy to get stained doors, cabinets and trim that match. Life is good.
So anyway, in my opinion, if the hospital is going to paint over stained wood, someone is going to have to do similar, and degrease/de-oil the wood first, sand it to rough it up, prime it and paint it, or it's just gonna peel off every time it gets scratched.