It all depends on where you are in line. Not everyone gets those paper FPs. At Everest, it's if you're inside the building main building (not just the part outside that's covered). If you're inside the building, you walk forward and get a FP, if you're outside the building you walk out the exit and get nothing. Every ride has a different cut-off point. For many of the rides, that cut-off point is merge. I'm fairly confident that's where it is on Seven Dwarves because a friend of mine was in line about 10 feet from merge when it broke down, and he said he didn't get a FP.
And as counter-intuitive as it is, if the ride was having that many issues, they might not give out as many paper FPs. If you have 2000 people waiting in line and the ride breaks down 4 times in one day, that would suddenly be 8000 people with extra FPs on top of all the FPs already distributed for the day. Think of the madness that would ensue every time the ride reopened. I once worked an E-ticket attraction that broke down 3 times on December 27th-ish (i.e., during the busiest week of the year). After the first down time, when we reopened, the FP was a 25 minute wait (due to all the original FPs plus the ones distributed to those kicked out of line and evacuated off the ride) with the line stretching waaaay out the door, and the stand-by line was two hours. After being open 90 minutes, we broke down again, and again had to kick everyone out of line. We had to turn away people who had waited those 90 minutes but were no where near the cut-off point and not give them any FPs. It sucked. We know it did. But we also knew that when we reopened, the FP line was only going to be worse than it was, and it was already at ridiculous levels. If we gave everyone FPs, the FP line would essentially be a stand-by line with a 90 minute wait. Sure enough, it was a mass stampede when we reopened that second time, to the point where there were people actually getting trampled on (and that was just to get into the FP line, which we always reopen first before the stand-by line). The third down time happened after I left, but I heard it was a nightmare. Again, it sucks in those situations, but there's really only so much you can do. It's a lose-lose no matter how you slice it.
I was in AK when Everest broke down on 6/3. I was going solo and circumnavigating the park clockwise from Africa. By the time I arrived at it, I was unaware it had been down for several hours. The FP+ line stretched all the way to Dino Land! Again, I didn't know that until after I rode it from the Single Rider line with only a 15 minute or so wait.