I think she means informed consent. My old GP said he would have prescribed it if I wanted to but he did monitor some light self medding I was doing.
A doctor with a DEA license can prescribe practically anything they want (with some limits) and hormones are not controlled narcotics like say Vicodin or similar. However, it's better to be under the care of someone who understands transgender care or at least hormone replacement therapy.
A month seems about standard. Not sure about the VA but I had to wait for a month for an appointment at Beth Israel in NY. Callen-Lorde is running 2 months and change now. So yea there is high demand.
You really don't need to see an endocrinologist. You can use a regular family doctor who is willing to prescribe. They will monitor your levels and standard stuff including glucose, lipids (cholesterol), liver and kidney function. They'll also take your blood pressure. All of this is basic medicine that is routinely practiced.