Games

They were also rare. To effectively pull off horse archery, you needed good horses, good riders that also happened to be good archers (both of which weren't trivial on their own, let alone combined) and good coordination. Bows are more effective the closer you are, so to get the most out of your arrows, you'll want to close in, but then you also need to wheel off again without your riders getting in each other's way, so you needed to drill maneuvers for that.

So you either need to have a sufficiently large body of soldiers with the leisure to train both archery and riding instead of working the fields, or you needed a society that treats them as basic skills anyway and only needed training in the military application. Nomadic peoples like the Scythians or Mongols often had the former, so they were notable sources of dangerous mounted archery, particularly where the raising and support of a professional army wasn't feasible. Rome had the Equites Sagitarii, but they were part of the distinct social class we would call Knights, so not your rank-and-file soldier (and those were already more professional than later levy- or retinue-based militaries).

So if we were concerned about accuracy*, these units should be expensive and require good management to make the most of them, but be very dangerous too. The point about open / closed terrain certainly fits as well.

What's a bit more foggy is how games usually handle bow effectiveness at range, but that's its own topic.

*I do care about accuracy, but not at any cost - games need to be fun too, and that's worth sacrificing some accuracy for.