Interesting idea. I cannot say that I can answer affirmatively nor negatively. There are also human elements to be considered. Humans are status-seeking social creatures. There will always be a stain of humanoid-delivered care, no matter how high-quality, as being not as high quality of all-human delivered care. This is, status accounts for a lot.
I can also draw pictures of how dangerous humanoid care can be, as there is a possibility in a break in the chain of responsibility. If a human medical professional messes up, you (or your survivors) can sue and seek damages directly, as well as sue the hospital and insurance system (with mixed results).
With humanoids? Currently, the bar is higher as the entity being sued is not the hospital, nor a person, or even a team. The only entities that can be addressed are the corporation the runs the hospital and the corporation that produced the humanoid. These two entities have an incredible out-sized advantage in terms of sheer delaying tactics, not to mention arbitration clauses and other legal innovations. Most injured will simply give up, which is a legal win for the two entities.
In my opinion, humanoid care will take a large amount of time, damage, and treasure to lower the costs. No actor will willingly give up their cash flow. My view may be too strong.