The point about heavy cavalry is that it relies on the close order charge and is an arm of decision.
I think that you may systematizing it : I'd agree if decisive field battles were common enough during Middle Ages (and even there, milites on foot rather than horse is well attested since the Xth*), but it's quite rare**, and rare enough to wonder about the "regular" usefulness of milites.
In siege warfare for exemple, which is the crushing majority of medieval warfare, the milites role is more to paralyze and weaken the adversary (chevauchées, raids, counter-attacks, preventing a counter-siege, etc.) and generally to threaten of weakening him structurally.
While you do have field battles, and with some of them forcing the decision and on which heavy cavalry unquestionably played a role; there aren't the norm : whole campaigns without such battles are quite current.
We're a bit fooled by the fact the chroniclers are nobles, issued from a milites (in the largest sense of the word) background and that naturally focus on it, and its "epicness" rather than everyday use.
But eventually, medieval heavy cavalry often had to undergo tactics that we'd consider part of dragoon or light cavalry in later periods.
Things would change from the XIVth century, with a more structured heavy cavalry in face of the new infantry (while, I stress it, infantry alreadu played a role by the High Middle Ages) that lead to a huge revival of heavy cavalry in a tactical purpose (even Italian armies used heavy cavalry as main military feature in the XVth).***
*Arguably, more in Germany and Italy than in France.
**Except, and that's worth mentioning, non-"feudal" armies. Which leads to pointing the ritualisation of the feudal war as well its paradoxal dependence/contradiction with castellisation of territories.
***The Caroline phase, though, points that such evolution wasn't bound to happen, especially if it went against military logic.
Although we have to be careful here as in Greek and early Roman warfare the aristocratic cavalry performed essentially screening and patrolling roles
It depends a lot about ressources and social order at hand, there as well : Macedonians and Thessalians had an important cavalry based on landed aristocracy; at the contrary of Greek poleis that had to count on a more limited wheat ressource, too limited to maintain a predominant heavy cavalry.
That said, yes. The differenciation between heavy cavalry as shock cavalry and light as skirmisher wasn't really a thing. (Unarmored lancers, for exemple).
Still : the point at hand there is that it's less of a cultural thing, than a social one (even if the former can play), and the semi-landed recruitement of Hellenistic armies allowed IMO the smooth transition to cataphract cavalry in Selucid Empire.
Combined arms was not alien to warfare of the period but recall light cavalry are essentially for scouting, screening, patrol, pursuit and raiding.
I'd want to stress that part of this (as raiding or pursuit) was more an heavy cavalry thing for most of Middle-Ages, while depending on the situation, light cavalries tended to slowly replace them by the Late MA.
Less because of structural issues, such as terrain (even if it may have played a role, but not decisive IMO), than the social role (ransom, "provocation" in order to force an open battle, unwillingness to mix with piétons, etc.)
it could even though this was more common later act as fairly effective heavy infantry once dismounted (you see this a lot on British warfare)
You already had such by the XIth century, especially among Germans (see the battle of Civitae, for example). I don't think it's really clear why it was seen as a German speciality, that early (it's more for England, when sustaining an heavy cavalry, critically on a more decentralized than desintegrated feudal structure, proven to be too great of a burden), but I'd think ressources problems (especially far from heartlands) played a role there (as it did during Crusades).
Its effectiveness, though, against heavy cavalry isn't that obvious to me, compared to its effectiveness was against regular heavy infantry. Before the XIVth, dismounted knights tend to be regularly beaten in such situation.
Giving that each aristocrat equipped himself he is as likely to try and be equipped for everything he can and the man at arms model thus predominated among the higher social orders in Western Europe.
Now one reason might have been purely social suggesting a simple change in culture could easily change that.
Rather than cultural, a social explanation may be more fitting : bows and crossbow were tought being non-noble weapons, if not outright cowardly (as in useful, but lacking elegance at best) : basically too useful to make them dismissed, but not "chivalric" enough (and there we enter class ethos) to be commandable.
This is hardly a western thing, at least when it comes to crossbow (Byzantine perception of the weapon as "diabolical" or even bows (you don't have a real mounted archery in the west of Arabo-Islamic civilisation).
I don't think the climate explanation is really fitting there, at least not directly : Byzantine mounted archery, for exemple, is directly influenced on nomadic warfare.
Conversly, it was present in Western medieval warfare in regions in contact with steppe warfare : Hungary (trough Pechenegs or directly); and if mounted bowmen wasn't that present elsewhere (a bit in Germany, IIRC), cranequiniers or mounted crossbowmen were.
Eventually, the main problems were their "ressource rivality" with cavalry; and a relative redundance with heavy and light cavalry anti-archer tactics. Still, it managed to be a thing for 2 centuries, which is not nothing.
Now if you want timariot siphai or allagion of cavalry raised by a system of pronoia or bow armed men at arms raised by feudal levy or even some kind of centralised state funded standing army then it is not impossible but they probably would not have been employed as light cavalry at least not so much in Western Europe.
It would actually bit quite hard : see, heavy cavalry social dominance really became a thing when the distribution of landed honores became widespread, early feudality explaining heavy cavalry, rather than the contrary.
The specialisation and shielding of cavalry came as a developement of an already established heavy infantry. To change that would require making archers the bulk and the main feature of Romano-Barbarian armies which I'm not sure you could even get, except with foreign "influence" more or less akin to nomadic warfare on Byzantines.
The absence of a true mounted (or else) archery threat at this point (unlike medieval Turks) allowed the bow perception as a pieton's and eventually non-noble weapon. (Which really comes far in western culture, and would require an early PoD to change that). Not unlike gunpowder weapons were seen in medieval Japanese warfare, while it allowed (and praised) bow by mounted elite.
And even that may not really deeply change things. You'd have more chances with develloping "native" light cavalry traditions (as long the OP is about light cavalry, and not siphai-like cavalry).