Given enough investment, Stirling engines could beat internal combustion for many applications due to a theoretical efficiency advantage. However, economies of scale and many decades of R&D give internal combustion such a huge advantage, that this probably will never happen.
At what point does the 20 year amortized cost of a system involving a space fountain or launch loop acting as the 1st stage of a vastly simplified and downscaled rocket break-even with some small multiple of the 20 year amortized cost of a space elevator? I suspect that this multiple might be small enough that economies of scale and optimizations of other systems could keep space elevators out of the picture forever.
ICEs gave us another boost in civilization, expanded the scale of our economies, which eventually led to enough R&D in electric engines that the latter are slowly but surely replacing ICEs now.
Maybe the same thing will happen here? The system you've described could open the Solar System for us, and time + advancements in in-space manufacturing could eventually lead to people building space elevators as an alternative.
(Also, Earth is not a good place to get the experience in building space elevators. The Moon is much better, and probably Mars would be a good candidate later on.)
At what point does the 20 year amortized cost of a system involving a space fountain or launch loop acting as the 1st stage of a vastly simplified and downscaled rocket break-even with some small multiple of the 20 year amortized cost of a space elevator? I suspect that this multiple might be small enough that economies of scale and optimizations of other systems could keep space elevators out of the picture forever.