First off it would make it a whole lot easier rather than skipping cutscenes multiple times. Second you can tell if it's a real run by looking at the mission finished whether it has your best time, grade and score. You are unable to get "your bests" on custom missions even if you keep doing the same one. Unless there is another reason for us to show or if I'm wrong then I see no reason not to switch to make it easier for retries.
I think the timer should be started at the true beginning instead of it just being when the 1st day starts, there is a lot in the beginning I believe would put more into this run. Also I think before each run, the Intended difficulty should be shown. Also if you want to keep the rules for Any%, I would like to hear your opinion and your ideas.