So, there's a lot of numbers there and I'm trying to keep track, but there's some significant flaws in your assumption.
1. There's nothing saying that all 57,000 people were even able to book fast passes. As you've said before, many had difficulty acquiring fast passes at the 60/30/day of mark. So that number is arbitrary and not likely to represent a real scenario. Also you're lumping in every single ride into the 171,000 fast passes which we know isn't correct because each ride has a different hourly capacity. So you're going to likely have more FP+ for PotC than you will Space Mountain.
The 171,000 has nothing to do with hourly capacity. It has to do with number of guests. We know that Magic Kingdom was able to accommodate 3 FPs per day for Magic Kingdom guests -- which is why they never had to implement a tier system at Magic Kingdom.
Average number of guests at Magic Kingdom = 57,000 X 3 = 171,000 FPs distributed on a given day. The actual number was much higher when you add in 4th and 5th FPs, but that was the number that might be pre-booked prior to park open, on an average day.
2. Again, your numbers seem off. Assuming that each ride has a capacity of 1200-1500 riders per hr, then you'd realistically be looking at 750 LL/FP opportunities per hr (assuming 50% G+ adoption and 1500/hr capacity) that puts your numbers at about 153k LL opportunities for all rides in a 12 hr day (again that's flawed because of the assumption that all rides have similar capacity).
No, the 1200-1500 is the number of "FP" riders per hour by my estimate, representing about 70% of total capacity, as was allocated under FP+. Many rides at Magic Kingdom have a full capacity of well over 2,000 per hour. For example, Pirates, Haunted Mansion and Small World all can handle over 2500 per hour. My estimates may be off slightly, but not by much. Some rides like Barnstormer, with low capacity, may only be able to distribute 500 FPs per hour, but something like Haunted Mansion can likely distribute over 2000 FPs per hour. So 1200-1500 is a fair estimate on average.
You can see an analysis here:
https://crooksinwdw.wordpress.com/2013/12/14/theoreticaloperational-hourly-ride-capacity-at-wdw/
3. Do I think every guest will book BTMRR at the beginning of the day? Depends on if that's the main attraction for G+ users. If it is, then more than likely it will. Same for why there's thousands of people trying to get a boarding group for RotR at park open at HS. There will always be a "must have" attraction that takes up all the FP/LL at the beginning of the day/hr. That's just how it works. That's human nature. If something is difficult to obtain, it must be good, if it is good, it must be obtained, which then makes it hard to obtain because of scarcity of resources.
So you're saying that Big Thunder Mountain is the most popular attraction by orders of magnitude over everything else? We know that's simply not true.
The "must have" attractions are being designated as paid-LL. That's likely designed to avoid the scenario you're proposing. Disney knows the demand for their attractions. They can insure that there is balance, by designate the "ultra high demand" attractions as paid LL attractions.
4. Moving forward with the pre-opening booking math, assuming 750 LL/FP per hr (50% adoption, 1500/hr ride capacity, 12 hr day) you're looking at around 9000 LL for BTMRR for the entirety of the day. Assuming 57,000 people are in attendance and let's say conservatively, 30% of those people purchased G+ for the day, that means 17,000 people will be wanting LL passes that day. Let's say even more conservatively, half of those people will be using G+ at 7 AM or park opening, that's 7,000 people getting LL at the beginning of the day. Under that wild assumption, after 7:15, you'd be lucky to get a LL/FP for any ride in the first 4 hrs of park opening. Let alone the possibility that BTMRR sells out in the first 2 hrs. If it doesn't, then that means you're hitting your phone every hr on the hr looking for that LL.
You're 750 G+/hour capacity is low. At a minimum, they will distribute 900-1,000 G+ per day. For a daily G+ capacity of approximately 11,000 to 12,000 per day. (BTMRR capacity is a bit lower than many other rides at MK, at about 1400-1500 per hour, but FP+ distributed 70% to FP, not 50%)
Now, let's go with your other assumptions: You assumed 17,000 people purchased G+. Ok. And ok, half of them use G+ pre-park open. That's 8,500 (not sure how you got 7,000 as half of 17,000).
Now, there are 17 G+ rides at Magic Kingdom. You really think BTMRR blows past every other ride in demand? 100% of people would choose BTMRR over the other 16 rides at MK?
Ok, let's assume BTMRR is by far, the most popular attraction at Magic Kingdom. Let's assume that nothing else is even close. In fact, for every 1 rider who wants to ride Splash Mountain, there are 5 riders for BTMRR! For every 1 Peter Pan fan, there are 10 BTMRR fans. Even under the most extreme scenario you can imagine, no more than 50% would book BTMRR as their first choice. (A good portion of guests don't even meet the minimum height requirement!)
So you have 8,500 people pre-booking in your math. Maybe 4,250 book BTMRR.
Which means... BTMRR would be booked up for 4 hours, before the park opens.
Now, under FP+, if you weren't lucky enough to pre-book, you would be stuck hitting re-fresh on your app, like you suggest.
But under G+.. there is no hitting refresh on the hour, as you suggest.
Even if BTMRR blows past every other attraction in demand, it means that at 9:01, you would open the app, and could book a 1:00pm return time for BTMRR.
Of course, under this worst case scenario, where half the guests are booking BTMRR, it means all the other rides would be wide open. So it means that at 9:01, the return times would look like this:
Big Thunder Mountain: 1:00 pm
Splash Mountain: 9:30 am
Peter Pan: 9:30 am
Haunted Mansion: 9:15 am
Pirates of the Caribbean: 9:15 a.m.
Under your nightmare scenario where everyone stampedes to BTMRR, there is still BTMRR afternoon availability, and you can get an instant G+ pass to every other popular attraction, since everyone in your scenario loves BTMRR so much.
In reality, it will be much more balanced. Maybe the more popular attractions at BTMRR see the 9:01am return time pushed to 11:00am, while a ride like Haunted Mansion sees a 10:00am return time.
In the end, it will work out to an average of 6-10 passes per G+ purchasing guest at Magic Kingdom over a course of touring for the day.