I concur - the first responders (Company aid men) were organic to the Rifle Company, with all additional litter carriers coming from the regimental or division's Service Company and the Battalion Aid Station, which was part of the Medical Battalion's compliment. Other units might be assigned bearer duty, including cooks, drivers, QM troops, etc. Medics assigned to the Med Bn were used to supplement the line medics if need be.
As I understand it, traditionally the aid men (medics) were the first to administer care and triage, the litter bearers evacuated to the Battalion aid station (100 to 1000' from the front line). A Regimental aid station could provide additional care or just supplement the other aid stations. The Collecting Company rounded up casualties from aid stations to the collection station (frequently 1000' back from the aid station) and to evac to a field hospital for treatment.
At any point the soldier could be treated and returned to the line, or allowed to rest pending evacuation or return to duty.
David J. Little
President, Tenth Mountain Division Living History Display Group
Secretary, Tenth Mountain Division Foundation
Board Member, Tenth Mountain Resource Center
Past Webmaster, National Association of the Tenth Mountain Division