What should the course leader do if some students were given less time for a quiz?
I believe there are a lot of circumstances that have to factor in to the decision, but to me, the most important initial detail is, do you have any way of reliably tracking whose time was cut short and who had the full time allotted? If not, it's going to be hard to analyze and correct the situation.

I'm assuming that, since it was a quiz, it's not a huge factor in their grade, so what I might do is either renormalize the grading or pro-rate it. Basically, if the quiz was supposed to take 60 minutes, and the most any students were shorted (by no fault of their own) was 10 minutes (still less than 20%), then the new baseline can be considered to be 50 minutes, or 83% of the original time.

To renormalize, multiply every student's grade by 1.2. For students over the new 100% mark, you could give extra credit or simply cap the score at 100%.

Or you can do a quick-and-dirty pro-rate by adding 17% to everybody's score, again, either up to a maximum of 100% or going over and giving extra credit.

I realize you have a huge class, which also implies you have a lot of early-undergrad students who might raise a bigger stink, but in my academic career, I've seen plenty of ex post facto shifting because of logistics, especially when a prof or the school is rolling out a new technological approach.