As per the linked website:
" If the airplane reaches a peak speed of only 10 knots beyond V 1, the brakes must now dissipate 20 percent more energy than had the abort been initiated at V 1.
Beyond the fact that there is no certification requirement for the brakes to be able to absorb any energy beyond that existing at the highest weight and V 1 combination demonstrated, there also is no performance data to know how much runway would be needed even if the brakes are able to handle the extra energy. Adding to the chaos, one brake must now absorb all the energy of the aircraft, as the blown tire’s brake has been rendered useless."
Basically you'd be unlikely to get the airplane to stop before the end of the runway because brakes are not designed to handle breaking at such speed (keep in mind that an airplane lands at a slower speed than it takes off thanks to the flaps).
The V1 decision speed is not fixed, it's calculated for each take-off based on weight, wind, runway conditions, maximum thrust settings etc.
It meets these constraints:
1. Low enough such that if you try to stop from that speed you will stop before the end of the runway
2. High enough such that if you have an engine failure at that speed, you will make it airborne on the other engine before the end of the runway
3. Not higher than rotation speed (you can't decide to abort after pitching the nose up and getting the aircraft airborne)
4. Not lower then minimum control speed (you can't keep directional control in case of an engine failure below minimum control speed, so your only option is to abort)
If there is no speed that meets all conditions, then your runway is too short and you can't go. Reducing weight helps, since you'll accelerate faster, stop easier, and take-off at a lower speed. So that's usually the solution if your runway isn't long enough.
Constraint 1 is why you're committed to takeoff above V1. If you would try to stop above V1 there is no guarantee that you'll stop before the end of the runway. While you are guaranteed to be able to take-off above that speed, even after an engine failure. So you take the problem into the air, run checks, and return.
