To answer the question as asked, no I've never heard of an airline "ignoring" V1.
To address the implicit question, I have no idea what the story is behind that IL-76 operation shown in the video, but I could creatively come up with all sorts of possible explanations for the takeoff roll that essentially matched the available runway: power not set correctly, takeoff data computed incorrectly, poor pilot technique, unexpected tailwind, aircraft loaded heavier than expected, or maybe that crew DID decide that it was so important to get home that they were willing to accept an absurd risk. But I wouldn't take ONE video without explanation of "what really happened there" as any indication of what airlines in general will do.
At its most basic, V1 is simply the point at which you transition from a "stop" mentality to a "go" mentality: if something goes wrong PRIOR to V1, we'll stop; if something goes wrong AFTER V1, then we'll go. Now, there are LOTS of things that complicate that discussion... not everything that can "go wrong" is the same: at 80 knots, getting an "anti-skid fail" light is probably not worth aborting for; at 80 knots, getting a loss of all generators probably IS worth aborting for (assuming that V1 is well above 80 knots). Just below V1, losing an engine IS something you'll stop for; losing one generator or one hydraulic system might not be. And this is all dependent on the aircraft... in the C-130, we were pretty stop-oriented, and we'd stop for all sorts of things immediately prior to V1 that in the 737 -- a much more go-oriented aircraft -- we'd take airborne. The difference being, you can stop the Herk pretty quickly & with a comparatively low risk of problems; a high-speed abort in the 737 (as in many jet aircraft) is much more of a big deal.
All that said, though, there is always a speed below which you'll stop, and above which you'll go. If an engine flames out at 20 knots, you'll stop in nearly any scenario I can think of (that doesn't involve bad guys storming the airport). In the Herk, it was common that your accelerate-stop speed was above your rotation speed, so V1 was essentially at rotation. But there was always some speed where your default choice between stop/go changed over.
You might "ignore V1" by simply saying that once takeoff power is set, we're going to continue regardless, but that's just bad decisionmaking by the crew, and not the sort of policy that I could envision ANY airline ever making. (Again, the case of mortar shells falling closer & closer to the aircraft & bad guys overrunning the airport might be an exception to this... take our chances with the ground rather than with the bayonets. But the outlandish nature of this exception illustrates how rare it would be to "ignore V1" in any considered scenario.)
Airlines tend to accept far fewer risks than other forms of aviation. The mission of an airline is to make money, and nothing interferes with that mission like the bad publicity & loss of assets (and potentially, certification) that comes from a crash. In the military, there can be a mindset of accepting lots of risks because troops are counting on us completing this mission. In general aviation, there tend to be individual decisionmakers, with little institutional guidance & corporate wisdom to back them up. But the airlines tend to be entirely willing to postpone a flight if it can't be performed safely, so for "airlines" to ignore V1 or other safety rules seems highly unlikely. (Which isn't to say that an individual crew might not do so, although at least at US carriers, they aren't likely to continue to be employed if they're found out.)
--- EDIT, added in response to updates/edits in the original question:
What you're asking is, in essence, can "they" get permission to take off knowing that takeoff roll = available runway, and if anything goes wrong after the point at which stopping takes all of the remaining runway, then they're dead. If takeoff roll = available runway, using all engines, after you've reached the speed at which you can no longer abort the takeoff within the remaining runway, and then you lose a motor, you now
- Don't have enough runway left to stop, but also
- You don't have enough runway left to accelerate to takeoff speed with the remaining thrust (less than the original case)
That's more or less the "bad guys storming the airfield" scenario, where you know that you're dead if you stay, so it makes sense to take off & hope that the motors keep running through the takeoff roll. Given that the risk of staying is so high, the risk of losing an engine during those seconds of the takeoff roll is the lesser of the two evils. And yes, in Viet Nam there certainly were times when C-130's would accept those sorts of risks to get the badly wounded out of dirt strips in combat.
Now, is that a reasonable thing to do in peacetime? Not in my simple mind. If we are assuming that the IL-76 crew did their performance computations correctly & their long roll was a known & accepted risk, then maybe they reached the opposite conclusion. It's not something that I'd expect an airline to ever approve with passengers on board, and we'd have to get creative to find a scenario where they'd approve it for a ferry.
Can a crew make that sort of choice on their own? Well, yeah, at least once. Would an airline approve that option? Hard to imagine.
