Because Dr. Phlox has the authority to order Captain Archer to sick bay, it seems strange that the concerned party chose to mutiny rather than exercising that provision.
However, Dr Phlox tried to quote Starfleet Regulation 104 section C, giving him authority to medically relieve Archer as unfit for duty. However, Archer threatens both Phlox and Trip with "confined to quarters" and they leave. Now they have no other option but to mutiny.
The writers gloss over the need for a "mutiny." The ship's doctor officially has authority to relieve the Captain of duty, and Archer's threat to have the doctor confined to quarters would be an empty one. There would be no reason to think the MACOs, including their commander Major Hayes, would refuse to follow protocol. The plot suggesting Archer manipulated Hayes and the other MACOs to be on his side--and Hayes admitting to Reed he likely would have taken Archer's side--is lazy writing that ignores military realities.
In effect, the writers have the Starfleet crew accept that their procedures are entirely flawed and there is no valid procedure for removing the Captain from duty no matter what may be wrong with him.
In effect, the writers have the Starfleet crew accept that their procedures are entirely flawed and there is no valid procedure for removing the Captain from duty no matter what may be wrong with him.