1943 Invasion
In 1943 the Allies would not have had overwhelming air superiority. But the beach defences weren't ready yet. Six of one, half a dozen of the other.
The primary problem for jet engines back in 1943 was that the alloys were refractory and oxidation resistant enough. They needed the maraging superalloys, and they didn't have them. They didn't even have titanium at the time.
It's a matter of hot creep strength that lets you engineer very close tolerances between the turbine blades and the cylinder walls. In internal combustion engines you have piston rings and oil seals. In a jet engine you have hot gasses blowing by, instead of through the rotors. The faster you spin them and the hotter they are, the more they stretch, until they touch...
Even ordinary alloys were in very short supply for Germany. Chrome, Nickel, Molybdenum, Vanadium, Manganese, Cobalt, Niobium, Tantalum, Tungsten...Hell, if even one of those German battleships had gone into imported ores stockpiles.