I had a door like that, had to lift the door and turn the knob counterclockwise only at the same and it worked fine. But it was MY door. I could do with it what I wanted. If the knob came off in my hand I had only myself to blame. But... I never tolerated such a door in any motel or hotel I ever stayed at and if my friends had such a door I would offer to help them repair it. What you own and what you rent are different. Don't know if you need a new door but fix the lock so there's no learning curve involved.
Tenants that have something to hide don't want to see you, but I want tenants who welcome my visit and appreciate that I care about where they live. I'm not judging how they live and I'm not checking up on them. Two or three days notice for a 20 minute with someone who wants to hear my problems and fix broken/loose stuff where I live? What's not to like? Give them a little coffee shop gift card and thank them for being a good tenant.
Final thought and not to sound paranoid but remember that house fires happen every day. Those loose wires on the range create a spark and someone is trapped inside or the first responders are delayed going through a futzy door and now your property is on the 6 o'clock news. Interviews with friends and family reveal you knew about the problems, believed these were exaggerated issues and did nothing. The heirs and their lawyer and your own insurance company will make your life hell.
Again, wishing you the best with this investment.