The answer is actually quite simple:
Airborne - use airspeed, it's the speed which keeps you aloft, makes you fast enough for take-off etc.
On ground (during landing or RTO) - use ground speed, as this is the speed used to determine the kinetic energy which needs to be absorbed from a moving object to one standing still. Remember the 1/2 * Mass * Vground squared, the latter being read from the GS indication?
During a stop, any stop, the object/aircraft requires a deceleration and this is done using (kinetic) energy dissipation, through use of various methods and factors. These include and are not limited to spoilers, flaps and airframe providing aerodynamic friction, brakes providing either a predetermined deceleration rate or a maximum pressure on brakes resulting in a stopping force, and last but definitely not least the reverse thrust.
If reverse is not used, the brakes needs to do more and as a result get hotter, thus needing more cooling afterwards, etc... Hence the two tables for cooling, with and without reverse in chapter PI.
Recommendation is thus to ALWAYS include GS in your stopping actions. This can be done by remembering the GS during RTO only as used in some companies. Or much more practical to call groundspeed out loud as a function of the "speedbrake up" call, eg "speedbrake up, 145". The latter would allow either pilot to hear the GS thus hopefully remembering it for potential cooling calculations afterwards. This should especially be done in airlines who operate short turnarounds. Its importance cannot be overestimated.
Airborne = IAS
Ground = GS