Yes, it doesn’t impact the probability distribution due to verifier. However, remember how you use PRNG and effectively due to the drafter is sampled from a different distribution initially, a separate rejection sampling won’t be able to recover what the “old PRNG” would choose in a “without drafter” case. Hence in my original post, it is about different trajectories you will end up with, not the correctness of each stochastic sampling.
If you assume that the RNG generates true randomness, then the two are identical. Only if you care about the determinism of the RNG (for example you want to use identical seeds and get the exact same generation between the two) it makes a real difference.