When you need 200 workers but only have 100, either the workers will have to do twice as much work in the same amount of time, or patient care will suffer.
I do not see a reason why a hospital can't use robots for some of that care.
If one of the tasks that needs to be done is someone going from room to room, collecting up empty water pitchers and replacing them with new full ones, does it matter if a caring human does it or a robot? I say let the robot do it...it's better than leaving patients thirsty while they wait on a worker that doesn't have enough time to get to it between other tasks.
If a worker can be assisted by a robot...the robot lifting a patient while a worker changes the sheets, it could allow them to make many more beds than the currently widely used method of rolling a patient to one side, making half the bed, then rolling the patient to the other side to make the other half, then rolling them back. And with a worker shortage, you need to minimize the risk of injury to your workers, otherwise the shortage problem is worse. Why risk the chance of a worker injuring themselves with pulled or strained muscles from moving patients, if you don't have to?
And how about letting the elderly stay at home longer? There are plenty that wouldn't need to go to hospitals so soon, if there was some way that would allow them to stay where they are. My grandmother would have loved to have a robot to be there when I wasn't, to pick up all the things she dropped on the floor and couldn't safely bend down to reach herself. (her dog was never too good at playing fetch)