The schools aren't going to look through a transcript update. Some don't accept updates at all, and those that do are unlikely to take "oh hey, I got proof that the F you either didn't notice or noticed and didn't care about because it was in HS really super doesn't count now" as a meaningful update.
No matter who you send an updated transcript to, anyone reading your app will still see the gpa you had when AMCAS verified you.
If you're going to change it, don't submit until you do. A change of several points on your gpa is probably worth waiting through July for, imo, unless perhaps your MCAT is a knockout.
Early-mid June is functionally equivalent to early July...you're still in that first batch or so of reviews. Mid-late July is early, but starting to bump into the peak of application season. Late July-early Aug is peak, and mid-late Aug is getting on the tail end of peak. Once you reach the end of August, you start to get into 'late' territory.
If you have solid stats and a strong app, timing is less important.
I'm applying super early because I have unbalanced stats and will look better when compared to far fewer applicants, so my MCAT can stand out.
For most people, it doesn't matter unless they're late.