Why go through all the trouble of using electricity as the heat source in a Brayton cycle engine instead of the obvious solution, using electric motors? Your statement that "we don't know how to go supersonic speeds with a fan or propeller" is clearly wrong, since supersonic aircraft have been flying with turbofans and no afterburners for almost five decades now.
At low-mach, the fuel burn is primarily producing torque which turns the fan. This fan could be turned by electricity.
At high-mach, the fuel burn is primarily producing pure thrust, and the fan portion hurts the engine's performance.
Supersonic turbofans are a compromise that have to produce thrust from takeoff through to high speed. I may be incorrect, but I don't believe we have ever built a ducted fan which can use pure torque to generate supersonic thrust, although I guess I don't see why it wouldn't be theoretically possible. We can put megawatts of energy into a fast-moving flow using thermal energy and a brayton cycle. In order to put megawatts of energy into fast-moving flow using rotors would require a totally new design - nothing like the supersonic turbofans currently in production.
No engine (of which I am aware) has air flowing past the fan blades at supersonic speeds, because the trans-sonic transitions tear engines apart. There is one prototype aircraft (a naval derivative of the F-84) which did have a supersonic propeller, but it was very inefficient and loud. All jet engines compress sub-Mach 1 flows, then expand them at the very back of the engine to produce supersonic exhaust.
Exactly. At high mach, almost all of the energy being put into the flow is from expanding combustion gases - not the compressor stage. If you wanted to go at supersonic speeds using only spinning fans, you will need a heck of a lot of fans. Maybe we could do it! But we haven't so far.
But part of a turbofan's thrust is from gasses from burning jet fuel and air expanded by the heat form that isn't it? Nobody has flown supersonic without burning fuel except in a dive, have they?
https://en.wikipedia.org/wiki/Fastest_propeller-driven_aircr...
So you can have an electric motor to power the fan, but what about the turbine part?