TL;DR:
- Why would the brewing process stop?
It doesn't.
- What is the role of the time spent before breaking?
There's brewing-related stuff happening during that time, and crust-related stuff happening during that time, and they're mostly not related. It isn't so much the "role" of the time spent up to that point, as it is various things are ongoing and at a somewhat arbitrary point you break up the crust.
- Why 4 minutes? What would happen with much shorter or much longer times?
Within reason, it wouldn't make much difference, at least that you would likely be able to tell. For cupping purposes, the more important aspect is that the procedure be consistent. It needs to happen at some point, and a time in the ballpark of 4 minutes is a good compromise to get the most benefit from the action.
Explanation
James Hoffmann does say in that video that you're stopping the brewing process, but I think that was an over-simplification. That great video is designed to simplify the process for beginners and explain the gist and some of the concepts involved. That statement isn't literally correct, but I'll explain what he's probably referring to.
What's going on with the brewing and what's going on with the crust are basically independent processes. Breaking up the crust doesn't really relate to the brewing, it's coincidental.
Brewing
If breaking up the crust stopped the brewing, that would suggest that most of the coffee grounds are in the crust and being extracted there. When you send them to the bottom, that stops the brewing.
It's the reverse. Only a small portion of the grounds are in the crust, they contribute little to the extraction while they're suspended in the foam, and the rest of the grounds are responsible for most of the extraction independent of what's happening in or to the crust.
There are two concurrent brewing-related processes that start when you pour the water into the cupping bowls: extraction of the coffee solubles from the grinds and settlement of the coffee particles to the bottom of the bowl. These both continue until you empty the bowl. For both, the bulk of it happens quickly and then tapers off.
Most of the extraction happens in the first few minutes. After that, the brew continues to get a little bit stronger and the flavor profile fills out a little more.
For settlement, the "big" stuff (normal-size grounds), falls to the bottom quickly, so it isn't a consideration, and it accounts for most of the coffee grounds. The fines (dust-size coffee particles), account for a tiny amount of the coffee, but it's important for those to settle to the bottom so you don't taste them during cupping (there's no filtration in this procedure). While the water is initially very hot, the fines don't do much settling. As the water cools down, they slowly settle to the bottom. By the time the brew temperature drops to a drinkable range, most of the fines have settled.
So brewing is the important thing going on in the first few minutes. When most of the extraction is done, that is about when most of the fines start to settle out. So you can think of it as a "brewing phase" then a "settlement phase".
The time normally selected to break up the crust roughly coincides with when the "brewing phase" can be thought to end, but that's just a way to conceptualize the process. There is no hard transition point, and both processes are ongoing. Breaking up the crust doesn't cause any change in the brewing.
The Crust
The crust is a combination of coffee foam, oils, and floating grounds. As water soaks into the coffee particles, it pushes out gas (mostly carbon dioxide), that collected after roasting. The gas rises to the surface, taking some lighter coffee particles with it.
Breaking up the crust might affect things in a subtle way. The crust might act as an insulating blanket. The brew loses heat through the cup, but the crust may reduce heat loss into the air. So it would keep the brewing temperature a little higher a little longer. To that extent, breaking up the crust would help the temperature drop a little faster, slowing down extraction, especially of flavors associated with over-extraction.
At some point, the crust needs to be broken up in order to get at the coffee for tasting. When's the best time to do it?
You want to give the floating stuff time to settle before you taste the brew. But you also want to allow enough time for all the gas to escape the grounds, and as much of it as possible to escape into the air so you aren't stirring it into the coffee when you break up the crust. And as mentioned, the timing might have a subtle affect on the flavor profile. The timing is a compromise. Doing it at about 4 minutes reduces the influence of technique and balances the various considerations.
Your observation about the insensitivity to the time is basically correct, although your range of 2 minutes to 8 minutes is probably more a reflection of how sensitive and trained your palate is, or what characteristics you were looking at. The cupping procedures I've seen range from about 3 minutes to 5 minutes, with 4 minutes being typical.
