Somebody once taught me that flying, in particular, is where Murphy's Laws are ignored at your peril...
.... If it can possibly go wrong - be ready for it to happen when you least expect it.
This is a great question and this thread shows the ambiguity that can arise. Sat at a PC, we've got oodles of time to discuss it too!!!
Shows what pprune can do!
I would like to add the Air France Concorde crash of Jul 2000 to the debate. A report I heard (assuming it was correct for the purposes of this discussion) said that the Concorde crew were informed of fire by ATC and responded that they knew, but had already committed to getting airborne. (Past V1?) Sadly, blessed with the benefit of hindsight, we know the result of that decision. If your decision allows you to walk away from a catastrophic failure such as this, then it could be argued (again, in hindsight) that your actions were correct. Ultimately, it might just have been the case that an abort, however hazardous, may have had a different outcome. Your problem may compound itself if you get airborne with a disintegrating tyre.
I am sure this crew handled the emergency they were faced with in an exemplary manner and may sadly have had never had a chance. I don't know all the facts.
However, I think V1 is an important event marker - nothing more. The Captain of the ac will ultimately have to make a go/no-go decision based on all the info he has available to him. A well trained and switched-on Pilot can assimilate a lot of info in a very short period. I guess we all hope for clear cut situations - of course life isn't always that generous.
In the scenario given, I think the 25 knots that is effectively all on the nose will help the abort case. (But not in terms of ease of control). Also, as mentioned, your climb gradient will be completely blown with your gear down. I'm sure the initial scenario had a 200' cloud base - so a visual circuit is out. So you'd have to tailor your decision to the surrounding terrain as well.
Finally, I hope we can all have these considerations worked out before the throttles are pushed up, and long before V1.....