To my best understanding, one the challenges in quantum computing right now lies on the quantum noise that affects the fidelity of the qubits to reliable execute calculations. Fault-tolerant quantum computings are going to be capable of correct logical qubits faster than the rate of errors that will arise on the computation.

To, address the question, I think NISQ based algorithms can still be useful in later-stage fault-tolerant quantum computers precisely because of the error-correcting capabilities.

Case in point: right now quantum simulation of molecules is really hot. As far as I know from reading papers, quantum chemistry problems represent opportunities to test quantum computing. Even with errors present on the computation, researchers have found that current NISQ-quantum hardware will still converge to classical known answers.

However, what if you find yourself with the following challenge: you would like to simulate the interaction of a pair of copper ions in a MOF to calculate the relaxation times as a way to compare with experimental findings using NMR techniques.This was a problem I was interested in for my MS thesis. Unfortunately, this right now, as best as I know from speaking with Dr. De Jong, is not feasible using a NISQ-based quantum computer.

However, in the future, quantum computers may be able to tackle such a problem.

So, to conclude, the algorithm will be useful. Perhaps it may no longer be suitable given change in the computing paradigm - from NISQ to fault-tolerant quantum computing. Perhaps new algorithms will be found on the fault-tolerant regime that can outperform current algorithms. But all in all, it's not the NISQ-based algorithm, but the error-correcting capabilities that are one of the issues at the moment.

I can only conclude that if error-correcting is perfected to the level required to reach that paradigm of computation, that current NISQ-based algorithms run on fault-tolerant quantum computers will outperform current runs on current hardware.

One of the issues, again, as I understand it, is the error-correcting capabilities we have today and their impact on current quantum computing.