I'm a long time reader of pprune and rarely get as tempted to post as this thread has made me, so I gave in.
SSG, it is obvious from your rantings that the only experience of operating big jets is that which you have gained from watching them take off from your own flight deck, or spotters fence.
I will confess to not knowing much about Gulfstreams, but I will hazard a guess that they are overpowered for their size, hence have reasonable acceleration even at high weights, and if operated from long runways will have not very many knots between V1 and VR. Well the larger jets operate differently, and you need to just accept this as fact!
There is no problem in rotating towards the end of a runway. The fact you seem to be missing (or deliberately ignoring) is that V1 could/would have been passed a long distance prior to the aircraft rotating. Therefore your points about the aircraft not having room to stop are irrelevant as a reject would not be attempted at this point unless the crew like going cross country on the deck!
Also, I'm wondering how the second youtube link you posted is in any way an example to prove your point?
This talk of flying to the fence and pulling back the stick is pure nonsense. If you have that little faith in how the larger jets are operated I am inclined to think you are not a pilot at all.
Back to the topic, I have been flying single aisle Airbus for nearly 2 years, so as an FO feel qualified to comment on what I would do in such a scenario.
If after a second, loud, speed callout there was no reponse, and we were on a short runway, V1 being close (time wise) to our standard 100kts call due to quick acceleration, then I would definitely continue. Assuming CAT1 or better weather then a quick radar vectored circuit would give time for the cabin crew to aid the Captain, the paramedics to be called to a stand, the situation to be thought through, and a nice long runway ahead of me. Another important consideration is that after a high speed abort on a short runway, the brakes will be very hot. Fuse plugs could melt etc and being stranded on the runway will just increase the amount of time it takes to get medical treatment to the capt. If an evacuation is required due to a brake fire then getting a heavy captain out of his/her seat in a hurry is not a task I would relish!
If, however, I was on a ridiculously long runway, Munich for example, where a typical V1 would be in the 140-150 region even at max flex, then by the time incap had been diagnosed the speed would still be way below V1 with maybe 2/3's of the runway remaining. In this case an RTO would not cook the brakes too badly, assuming I would have the awareness to disconnect the autobrakes and reduce the deceleration.
Of course being in a bus with a tiller on the RHS makes this scenario better for an incap captain and definitely leads to a better outcome in the eyes of the airport operater who doesn't have inconvenience of a blocked runway.