Nedorim 39-40 that praises bikur cholim seems to describe both benefits – spiritual (people davening for the sick person) and material (R Akiva visit leading to better health care). As is often the case, Gemora does not seem to argue one against the other but asks us to keep both in mind. As part of the discussion, there are statements about what plants are beneficial and how one would recommend a cure to the other. It seems to be assumed that people who can cure someone else should.
> because the secular world respects doctors so much
I am not fixated on titles. Dr. Jill Biden is not relevant. It is just that a heart surgeon saves thousands of lives, way more than an average Yid who is diligent at bikur cholim. Should we suspect him of all the bad things that you cited, or stam show him the respect?! Same goes for someone who came up with a new medicine, or invented seatbelts. I don’t think we show enough respect to such people. We tend to think that “they are doing their jobs”.
bottom line – what are difference in schar between people doing mitzvas for free or as part of their job?