Why don't Jesus and the Apostles ever recommend going to a physician? They certainly do not get around to each and every individual, and there are no claims of curing an entire group of people. Are they entirely ignorant or have no faith in ancient medicine? Such a suggestion is included in the Book of Sirach, Ecclesiasticus 38: 1-15. Also in 2 Chronicles 16:12, King Asa resorted to physicians for gangrene. In Mishnah Kiddushin 4:14, the best among physicians are destined for Gehenna.
We can only speculate about why Jesus or others did not do something, for the Gospels don't reveal much of their motives or reasoning. I don't see signs of particular ignorance/superstition on their part, but there could have been some distrust or skepticism toward those in their world who presented themselves as physicians. At least, the way Mark's Gospel describes a woman suffering from a hemorrhage (Mark 5:26) suggests that the author of that Gospel might have been distrustful of doctors (compare the less severe statement in Luke 8:43). The Gospels portray Jesus as aware of healers/physicians (see Mark 2:17), but apparently not interested in making referrals. As your references to different ancient Jewish texts indicate, there were a range of opinions toward doctors. Some sources from near the first century refer to doctors as expensive and so only accessible to some people. Other sources describe medical techniques that we'd regard as horrifying (and "ignorant," from our view). The miracles stories in the Gospels have, it seems to me, an overarching aim that is less interested in criticizing ancient physicians and more interested in proclaiming God as the only truly effective healer. Jesus, in fact, does display an ability to heal groups in one fell swoop (as in Luke 17:11-14), but the Gospels don't draw much attention to that. The Gospels nevertheless depict him performing healings without techniques and at a scale that "an ordinary physician" in his day wouldn't be able to match. Perhaps in the view of the Gospels Jesus makes those physicians unnecessary. Don't take this as medical advice for today!
It must be remembered that however effective ancient doctors and/or medicine may have been, it was always God who was the ultimate source of effective healing, whatever form that healing may have taken.