I wonder if the impetuosity of Royalist horse has become a bit of a wargamers' trope? The two most written about battles of the first civil war are Edgehill and Naseby, both of which featured Rupert commanding the right wing of horse, and both of which saw Rupert victorious, and that wing of horse pursuing their broken enemies from the field.
However, I wonder if using just these two instances as the evidence of Royalist horse always rashly pursuing, is itself, a bit rash ;-) .
The need for horse to pursue a broken enemy was standard practice before, during and after this period. It was necessary to ensure the enemy did not reform. The desire to loot and pillage was also not only a temptation for Royalist horse.
At Edgehill I think it is fair that the Royalist horse's pursuit was rash and was due to inexperience. The second line of Rupert's right wing should not have followed the first in pursuit. A mistake by the inexperience of the command. At Naseby the situation was very different. Rupert's right wing at Naseby was significantly outnumbered by Ireton. It was necessary for the Royalist second line to be committed to break Ireton so there was no fresh second line to turn on the New Model foot. Cromwell on the New Model right had adequate numbers to break Langdale and still have a reserve to exploit his victory.
Despite this, I suspect that our the wargamer's "truth" is now set in tablets of stone and Royalist will forever be 'A' class looneys with flowing locks and feathers in their hats, and their Puritan adversaries will be dour men with looped sleeved coats, and of iron wills.
[It is worth noting that in the very good 'For King & Parliament' rules, by Messrs Miller and Brentnall, all Horse (and Highlanders!) pursue broken enemies.]