If I may add a couple of observations to PantLoad's commentary...
(a) if one looks at the climb performance, say, gradient against speed, then you end up with something approximating an upturned teacup section. That is, at the low speed end (assuming you are not truncating the graph due min speed considerations) you will have a relatively low gradient. Then, as you increase speed, the gradient increases. Further increases will get you to a plateau region of the graph for which significant speed variation doesn't result in much change but, somewhere in the scatter, will be the maximum gradient. Beyond this, increases in speed see a reduction in gradient.
The routine certification V2 is somewhere down at the lower speed (and lower climb gradient) end of the graph for a bunch of reasons.
If you have sufficient runway etc., then you have the option of increasing the speed (within a reasonable range, perhaps 20-30 kt) to pick up some extra gradient capability at the expense of a quite significant runway distance penalty (keep in mind that distance needed considers speed squared as a first approximation).
There is no useful purpose served in going further up the speed scale as the distance penalties become rather horrendous and you rapidly start seeing the gradient benefit fall away .. ie a simple case of cost/benefit.
As PantLoad suggests, if the runway/obstacle geometry suits (and it doesn't always conveniently do so) a takeoff scheduled with a higher V2 (overspeed takeoff or improved performance takeoff) you might be able to get a better TOW as limited by obstacles.
(b) a combination of lower flap and overspeed schedules may be needed to obtain the high end of the MTOW capability for an aircraft .. ie you have to have very long runways to use every last kilo capability.