Aira, please cite where Rav Moshe’s alleged teshuva appears. I find it very difficult to believe that he would write something like that. Besides, doctors and nurses can certainly find professions that pay at least equally well, without the crazy hours and astronomical malpractice insurance premiums caused by juries who think that they are doing chesed by giving out huge awards.
In any case, Chazal say that it is a mitzva (Sukka 25a). See Nishmat Avraham, OC 38,6 in the name of Rav Yitzchak Zilberstein that this is even if they are paid.