When I was there in April, it was a few seconds after my phone clock hit 7:17. (The case for RoTR, FOP and 7DMT - I did not use second drop for Remy.) Since my trip, I have seen reports (mostly on Genie Usage and Strategy thread) of the drops occurring a little later (7:20?). I suggest going on MDE in the days leading up to your trip and checking for the drop times each morning. If I hadn't seen the drops with my own eyes during the week prior to travel, I wouldn't have been confident that waiting was ok.
Signs that there is a drop of many openings vs a handful of released times (as people have issues with payment and whatever else) is that a major drop will send the available ILL window times back to early in the park day (and they will stay there even if you refresh) and the time won't have changed drastically when you see it on the payment page (you can only get that far on your park day).
I started looked around 715 the first day (it ended up being down a good chunk of the day and we had plans somewhere else so guest services gave me a refund). The second day I bought one, I think times started appearing again around 7:10?