When you clear a stage for the first time, you have to wait for 9 seconds before you're back in the world hub. But after the first clear, you can quick retry after hitting the goal, then pause and exit to hub, which only takes 3-4 seconds, saving around 4:00 to 4:40 across the whole run based on execution.
In mirukono's New Game 1.01 run, they close the game after hitting the last goal of each world to skip the cutscene, and the game detects that they completed the stage without waiting the 9 seconds (Steam leaderboards also add your time as soon as you hit the goal).
So in a New Game+ run, would it be allowed to quick retry and exit after each stage? I personally see no reason why it shouldn't, and it would also set it apart from New Game more. You could just make it a rule that if you retry too quickly and don't actually clear the stage, you need to redo it or else the run doesn't count (menuing this fast shouldn't be necessary before reaching a high level).
Yeah, it should be fine to skip the stage clear screen with the quick retry.
Personally I have been refraining from using this trick in my own runs to keep things casual, but I agree it would further distinguish the categories.
As long as the moderator can visually confirm the crystal has been grabbed, it's all good.