It's possible Neoshamanists monopolized healthcare in the same way that the Bright Way monopolized other industries. One of the criticisms leveled at the Bright Way during the Shakeoff may have been that research monks were so preoccupied with what was above, ignoring the earth under their paws.

In any case, the Nonagentivist branch of Neoshamanism bore much fruit in the sciences of biology and ecology, and by extension medicine. The Bright Way may have had an excellent understanding of anatomy, but things like microbiology, epidemiology, and pharmacology were afforded less attention.

However, the Bright Way began integrating these seemingly unrelated fields of study into their pursuit of the Great Commandment, and especially when they bore fruit in the form of successful orbital flights, research monasteries began casting a much wider net, now understanding that knowledge of seemingly unrelated fields was crucial in accomplishing their goals.

This is why Neoshamanism seems to have a burst of scientific output immediately after the Shakeoff, especially among the Mindseekers and Lifebringers, but tapers off soon after the space age begins. The Bright Way had much more resources, and began eating the Neoshamanists' academic lunch once they realized “Oh hey this other stuff is also important.”

I'm pretty sure the Neoshamanists started the custom of healers going bald, as they were the ones to discover the importance of hygiene in medicine.