The flow of heat from Earth's interior to the surface is estimated at 47 terawatts and comes from two main sources in roughly equal amounts: the radiogenic heat produced by the radioactive decay of isotopes in the mantle and crust, and the primordial heat left over from the formation of the Earth.
Earth's internal heat powers most geological processes and drives plate tectonics. Despite its geological significance, this heat energy coming from Earth's interior is actually only 0.03% of Earth's total energy budget at the surface, which is dominated by 173,000 TW of incoming solar radiation
It seems to me that building space based a solar array even a half the area of the earth is going to get you significantly more energy than geothermal radiation from earth. You could as you say mine the core for heat directly but then you are still going to have to radiate off all that heat (you cant get work from there just being heat: it has to flow) which might mean building what amounts to planetary scale heat sink to radiate it all into space. Even with a nuclear power station the bulk of the thing is not where the heat is generated but in the cooling system which might involve taking in a river and releasing a slightly warmer river. I would say your geothermal plant becomes a world engine when it becomes no longer viable to release its waste heat into the environment (because you will boil the oceans and heat the atmosphere) and instead you have to build that massive space based heat sink.
My guess is that humanity will just use all sources of energy available so long as there is a positive return. I think you downplay fusion a little too much. Think about setting up floating fusion stations on jupiter and beaming the energy to wherever you need it. There is so much hydrogen there that I doubt you will burn through it all before all the planets in the solar system have cooled solid and you know the technology will be usable in the next solar system because the hydrogen in gas giants is not being used up while the heat of planets is.