If you uncheck the "Trigger initially on" and create a "Run trigger" action for every single trigger of your map in a "Map initialization" trigger, would it reduce the loading time?

The map still has to load everything, it wouldn't matter how you set it up. If you added 'Run Trigger' for everything, it'd probably just make it take longer, as that's more stuff to load.

Or it would load and then lag the first few moments of the map running all the triggers.

What is that "Don't preload triggers" option we can check from within the Editor?

I'm unaware of such a thing.

Map -> Map Options...
There's the "Disable Trigger Preloading" box you can check.