The Prefecture costs 2500 and will only pay for itself after 5 days, i.e., by day 8, you will have only gained 1000, while being at a loss in terms of army, tempo, and everything else that an additional hero could have gathered, which was either not purchased or purchased too late.
The Prefecture will pay for itself in 3 days (even with a margin), not 5. Therefore, the rest of your statement does not contain any "usefulness".
