The statement that losing one engine while both are under full power will render the aircraft uncontrlable is incorrect. With no other damage it will fly fine. In fact, on modern aircraft the autopilot even takes care of the yaw.

All Boeing airpline being able to fly with engine loss at V1 is correct for all their transport planes. In fact, all Part 121 certified aircraft are required to be able to do that at max gross weight. Part 121 certification is required for all aircraft approved for service with over 19 passengers. Yes, even most those prop powered commuter planes will do it.

Take off is only aborted BEFORE reaching V1 (otherwise known as commitment speed), V2 is the speed the aircraft is at when the main wheels leave the ground, Vr is the speed the nose starts being picked up (rotation speed). A demonsrated V1 "cut" is required at certification for all Part 121 aircraft and pilots have to demonstrate their ability to fly out of a V1 cut during type certification and usually during recurrent training (every six months).

And finally, it is true that a three engine plane may fly on only one engine, but it is doubtful that it could climb or go around on approach even under ideal conditions. But one will get you safely to a nearby airport unless there's a mountainess obstruction in the way.