Couple of thoughts -
(a) There aren't many turboprops that don't have the power to either climb away well, or speed up when very light
Same applies to the jet brigade as well. However, the problem is not acceleration capability but risk of yaw departure. This is mainly a worry for min speed schedule takeoffs at low weights for aircraft which are Vmcg/Vmca limited. Hiccup once and you are in a world of hurt.
(b) The speed at which the takeoff must be continued, if the abort has not already been commenced.
This has been the mantra for the past 25-30 years. To paraphrase .. if you're not already stopping by V1, then you're going .. (caveat is those oddball loss of control situations when one might be forced to abort above V1)
As Brian notes above, it is important to keep in mind that V1 has varied in definition over the years and the relevant words in your particular AFM are the best guidance as to what is expected for the performance data to be reasonably predictable.