I think that society DOES value what you mention. Look at the high pay that doctors get. Also, although many teachers complain of low pay, they receive high psychic income, as well as schedule that allows for more free time than most other professions.
Can't pay your bills with psychic income and for a long time most doctors were men. I think being a doctor frequently has more to do with abstract things such as reading results and making diagnoses rather than patient care. The actual patient care, provided by nurses, orderlies, etc, pays much less, although personally I think it might be of greater value to the patient.
It is much simplier than what you state.
"Barriers to entry" are the reason for the pay discrepancy between doctors and teachers.
Anybody can be a teacher, subsequently there is a large supply of potential teachers willing to replace current teachers who are not willing to teach at the prevailing wage rate.
The converse is true for doctors.