Cavalry used up horses the same as infantry used up shoes. The Union army paid $130.00 per horse. Each horse consumed 26 pounds of fodder every day. Horses had to be shod on a regular basis. On campaign, each trooper would need two, three, four or more remounts. That is $520 per trooper; $20-30,000 in 2020 dollars(+/-).
A horse had to be five years old to withstand the rigors of military life. That means that every horse that came of useful age was the issue of a stallion that stood to a mare before 1859-60. From the first shot that was fired, horses were a diminishing asset. There was absolutely no mechanism for ramping up production for the war.
An excellent professional essay on the Union remount system is in the Ten Volume Photographic History of the Civil War, The Cavalry. It is available online. The photos & cost figures will answer your questions in detail.
When you consider that an infantryman was paid $156 per annum, a carpenter/skilled craftsman $300 per annum, the $130 cost of a single horse comes into perspective. In 2020 dollars, the 120 horses of an artillery battery were worth $750,000 (+/-). A mounted battery would more than double that figure.
To put the cost of feeding horses & mules into perspective, during the Tullahoma Campaign that began June 1863, the Nashville & Chattanooga RR was used exclusively to ship fodder southward. Food & ammo were carried on wagons.
The simple answer to the question is that a cavalry regiment cost many, many times that of an infantry regiment to raise & maintain.