TL;DR: Yeah, but not completely. It is also inadvisable, unless budgetary or other extraneous factors play a part in decision making.
This would only create soldiers relatively immune to prolonged marches or prolonged operations and not able to replace some frankly inhuman military capabilities.
For an example: None of us can go supersonic naturally and the gearing needed to drive a ducted fan to push a plane to be supersonic would be insanely heavy because of the complexity and forces that a human driven supersonic engine drive train would require. For a more pedestrian aeronautic capability, the Merlin engine of the spitfire is roughly one thousand horsepower. A good cyclist could probably manage 1000W or about 1.35 horsepower. For your superhuman to manage a spitfire level of performance, they would need to be roughly 740x stronger than the normal human being (not to mention the caloric consumption that would entail)
That being said, it would open the opportunity for better infantry logistics when working in small groups, they can carry more resources and therefore fight for longer, not to mention that broken ability to not need sleep.
It could also open up some niche capability though, like bicycle powered transport gliders for short ranged hops between neighboring camps/bases. forward dirt airstrips could be used to resupply using those gliders especially when used like an air tow fashion like the soviets had done. (when some cargo planes might get a engine full of dirt trying to RTB, that is)
(the process would probably be that the mother plane would tow the gliders in a route that goes over multiple forward bases at once and unhook the individual gliders for them to land at their respective locations. Then once unloaded the glider would limp back to base or the logistics center they were at under the power of a couple of these super bros on a tandem bike to return the glider.)
Other forms of transport could be flatbed trikes, all terrain pedicabs for long endurance marches (one guy pulls four guys, rotate the guys every unit of distance, one guy could have a mounted machine gun lol).
On rail, this could be used in conjunction with slip coaches to add a margin of safety and insurance (if the slipped coach was going uphill or stopped early, the guy can pedal it into station) and many other light transport.
This will free up the military to focus on heavy equipment and not light trucks and whatnot; the only light vehicles would then be government official cars and various vehicles that has mounted on it some appliance that needed external power like SAM trucks, ICBM trucks, Radar trucks, Kitchen trucks (maybe), Long Range Communications cars, etc. Pure light transport trucks could be de-emphasized in favor of more heavy transport trucks as well.
Conceivably, this pedal power could also be used in the face of oil shortages due to blockade or prolonged war.
IIRC armies back in the day also experimented with bicycle corps to see if simply upgrading with cheap bikes could provide a big enough mobility and carrying capacity boost to infantry. It seemed it was born of a necessity to replace horses rather than motorcycles (horses were relatively harder to support, logistically). Though, sidecar bicycles are a thing -- out of which necessity is beyond me.
Keep in mind that this is if your country's economy is relatively poor, that is, if the economy could not purchase and support proper mechanized transport on such a scale that it might need to. It is much more efficient to let a specialty engineered part do the job than if a human were to waste energy doing one of many things that it could be doing otherwise. Its a waste of intelligence and quite the waste of energy for a human to perform transport under their own power.