only increases the initial climb if V2is less than max V2(around 1.35VS).
You will get a better gradient for quite an increase in V2 at the expense of field distance.
However, that is not the point in question.
The main problems with an ad hoc overspeed takeoff is that
(a) the flight path will be under that calculated from lift off until the improved higher gradient intersects the calculated gradient. It could be a case of there be dragons in that region.
(b) the higher speed results in larger turn radius if the departure involves turns to manage obstacles. This also tends to result in the subsequent flight path being sufficiently displaced laterally from that calculated to put the aircraft out of the calculated obstacle splay region. Again, it could be a case of there be dragons