It is theoretically a good idea, but as all good ideas, it usually experiences a stark reality check when someone asks 'who will be paying for this?'. It is true that in the High-to-Late Middle Ages (10th-15th century), the cavalry rose to prominence, especially when we're talking about the heavy cavalry ubiquitous across Europe, but its numbers were limited by many cultural, political and economic factors. It should be noted however, that the latter varied between regions, as e.g. Mongols or, to large extent, also Poles and Lithuanians were primarily using cavalry well into 17th century.

Now, the costs notwithstanding, the one thing that made availability of cavalry quite limited was the sheer physical and mental requirements for such service. Each heavy cavalryman had to know how to ride the horse in the first place, fight on horseback with various weapons, from lance to axe to sword, how to steer the horse in combat and how to stay in formation during the attack and retreat. These skills required quite extensive training that nobles, being raised among warriors and having a lot of spare time were able to practice from the early years in mock combat, common pastimes such as hunting and, last by not least, by socializing with people possessing actual combat experience. The number of people who had the time, means and opportunity to learn all of this was naturally limited and so was the capacity of forming the cavalry units.

The physical properties were by no means less important and contrary to the situation today, where people tend to be more or less uniform in terms of size and strength, until 20th century the differences between classes could have been tremendous, as poorer people were intermittently or even almost constantly undernourished (both quantitatively and qualitatively) what resulted in significantly smaller stature and strength that were directly linked to the lower combat prowess in comparison with well-fed nobles. I touch on the subject here, although the data invoked there do not relate to Middle Ages, but they may shed some light on the matter.

The cost of the mount was also far from a trifle matter, as it usually was pretty hefty with the caveat that the horse could have been easily wounded in battle and being surprisingly fragile animals, wounded horses were not much useful after sustaining a serious injury. This is why most cavalrymen usually had more than one charger (a horse specifically trained for combat), not necessarily much sought destriers, but well-bred and trained mounts nevertheless. Sources vary, but it is assumed that each noble combatant had to have at least 1 or 2 spare combat horses, with additional spare horse for his retainers (a retinue of 5 people was assumed to have 12-15 horses, not counting draft horses for wagons). The prices in the Middle Ages are quite finicky due to scarcity of records, large fluctuations and differences between countries or regions, but e.g. Dyer states that the average horse bought by a knight in 1374 had a cost of £5 (it was definitely not a draft horse, as the same source states their price as 10-20d, i.e. 5-10 times lower), but it also mentions that exceptional destriers could have reached and exorbitant price of £80 and more.

How does compare to cost of other equipments? Dyer [1] states that the 'total armor owned by a knight' was worth £16 6/8 in 1374 (although it is not sure whether 'total armor' meant a single suit or more) while Storey [2] lists the average price of the full suit of armour between 1294-1339 as £1 17s. Add a lance for 8d and sword for a 3s 4d and you could have armed a knight or man-at -arms for slightly over £2. This means that the cost of weapons and armour was much lower than that of even a single combat horse. This was carried to 16th and 17th century, when the cost of weapons and armour was lower due to developments in technology and ability to maintain intensive production in areas specializing in manufacture of such goods, as described in this thread by u/WARitter.

Summing this up, cavalry was much, much more expensive to field and the number of people physically able to effectively fight on horseback was relatively small.

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[1] Christopher Dyer, Standards of Living in the Later Middle Ages, Cambridge University Press, 1989

[2] Randall Storey, Technology and Military Policy in Medieval England, c.1250-1350, University of Reading, 2003