Should we replace the trained router with a GPT-4 promp...

Using an LLM to route is often counterproductive because the cost of the LLM call may exceed the savings gained by routing to a cheaper path. Additionally, you lose the calibrated score (τ) needed for thresholding against cost ratios. Finally, the failure modes of the router and the generator become correlated, meaning the system is more likely to fail on the same types of queries. The only scenario where this is viable is if traffic is genuinely open-domain and the branches are significantly more expensive than the routing call itself.