He is fully aware of the challenges.
I think most Americans have this idea that the United States military is like a fire extinguisher mounted in a glass cabinet with the words "in case of emergency break glass" painted on it. The fire extinguisher can work, but it is not designed to put out a forest fire-sized pandemic.
See this realistic evaluation of the problems that will be facing us as we move to provide vaccinations to Americans: "I Was a Military COVID Planner. The Vaccine Rollout Is Going to Be a Nightmare."
Resources are constrained, even in the military. When you talk about priority inoculations for medical personnel and first responders, in the military it also means the entire fighting force needs to be protected in order to be able to project our military power (hopefully not in Iran) wherever it is needed. Remember the aircraft carrier coronavirus cases from earlier in the year?
As Kris Alexander rightly points out, it is the medical system here in the states that is woefully unprepared to care for sick people in normal circumstances and now [my thinking] is being totally overwhelmed with the current surge. Today, Wednesday, we just passed 250,000 deaths from COVID-19. Yet some people think that COVID-19 is still a hoax, even when they are on their deathbed.