It can cause some fatigue, sure, but much of that fatigue will come from a combination of general immune response and tissue repair. Symptoms can appear modest even as your body is waging a very costly war internally.
Right, it makes sense that if a great deal of energy is going into immune response, the rest of the body can get shorted.
The converse is also true - being “run-down” weakens immune response. I’ve let that happen to me too many times... I have little doubt this is a problem when hotspots overburden health care staff.