The expense of shipping horses overseas pushed for more infantry instead, and reliance on provincial horse regiments instead. The 16th and 17th light dragoons would be the only regular establishment horse regiments sent overseas. British leaders hoped that they could raise cavalry in a way they could with infantry. In the Southern Campaign, leaders also realized how ineffective most of the tactics would be due to the terrain. They learned this from their experience in the French and Indian War when they also did not use a lot of cavalry, if any (I think only one regiment of horse fought in that war). With their lack of cavalry, the British army employed different tactics with their infantry. No longer in tight formations but in open orders and moving quicker.
The British army did have some provincial cavalry regiments under their belt. The British Legion, Black Dragoons, Queen's Rangers, and Virginia Light Dragoons are a few of the horse regiments employed. It was the availability of horses that also led to fewer horse regiments used. However, it did not mean that they did not have an important role in battles. The army used them to hit the flanks or weakened groups. Rarely, did mounted troops have success with full frontal assaults as it meant the enemy soldiers would have some preparations against charges. Both Simcoe and Tarleton suffered casualties when their cavalry charged against infantry head on. It was a gamble to strike head on, and like soldiers, horses take resources to replace and train. With the change in tactics: open orders, two ranks, and heavy reliance on bayonets for infantry - the cavalry became more of an additional role than the main one employed decades before.
The role of dragoon and light cavalry differed from the heavy cavalry used in the past. Heavy cavalry featured heavily armed troopers with protective armor such as a cuirass and sword. Heavy cavalry can be considered the mounted shock troop in comparison the infantry's grenadier. In that sense of comparison, the light cavalry would be the horse counterpart to light infantry. They had a role of skirmishing and harassing troops to break them. Dragoons in the American War had a dual role of not only being mounted infantry but also light cavalry. Dragoons armed with pistol or carbine could close in fast and fire, often the dragoon would use the horse to get close and dismount to fight. Firing from the ground meant a more accurate shot as it would be difficult to fire from horseback with constant movement and jostling.
The weaponry of the field also affected tactics. Cavalry had three choices: the sword, the pistol, or the carbine. The sword strikes the image of the traditional cavalier charge but meant the troops faced close quarter combat. Pistol and carbines created the role of mounted troop but lost the advantage of hand to hand fighting. A dragoon had one chance to hit an enemy before needing to reload, and that opened them up to enemy attack if they had not been routed. Even with a one-handed weapon such as the pistol, it would have taken far too much time to draw the sword out after missing a shot. Cavalry found its best utilization with cold steel, as the trooper could break through the ranks and continue chasing down any enemy soldiers that routed.
The British army used cavalry at many of the important battles of the Southern campaign and some of the smaller ones, most notably the British Legion with the attached 17th light dragoons. As mentioned before, terrain dictated the battlefield tactics used and with little open field - cavalry could not often charge a full frontal assault. Tarleton suffered heavy losses when he charged the militia at Cowpens, who had cover and resolve against the charging troopers. He had better success at Camden, several months earlier, when he attacked the rear of the Continental army and routed them. However, the success had depended on the effective fighting of the infantry rather than other cavalry units.
The availability of horses also proved difficult. Not to mention the army still needed horses for equipment and transportation besides dragoons. The resources to maintain them as well. Most dragoon regiments had their own armorer but it still meant that horses had to be taken off duty.
Daniel Murphy, "Infantry v Cavalry," allthingsliberty.com.
Matthew Spring, With Zeal and Bayonets Only.
In fairness, the British did use some cavalry during the war. The first British cavalry to arrive in America were the 17th Dragoons, who appeared in late 1776. They quickly made an impression on Washington, who ordered:
White Plains, 27 October, 1776
"The General observing that the Army seems unacquainted with the Enemy’s Horse; and that when any parties meet with them, they do not oppose them with the same Alacrity which they shew in other cases; thinks it necessary to inform the officers and soldiers, that, in such a broken Country, full of Stone-Walls, there is no Enemy more to be despised, as they cannot leave the road; So that any party attacking them may be always sure of doing it to advantage, by taking post in the Woods by the Roads, or along the stone-walls, where they will not venture to follow them; And as an encouragement to any brave parties, who will endeavour to surprise some of them, the General offers 100 Dollars, for every Trooper, with his Horse and Accoutrements, which shall be brought in, and so in proportion for any part, to be divided according to the Rank and pay of the party.
General George Washington"
On December 11th, 1776, after having been harried by British cavalry for months, Washington wrote to the Continental Congress:
"From the experience I have had in this campaign of the utility of Horse, I am convinced there is no carrying on the war without them and I would therefore recommend the establishment of one or more Corps . . . In addition to those already raised in Virginia."
So, British cavalry clearly had an effect on the war.
However, the British never deployed massed cavalry during the war. Only the 16th Regiment of (Light) Dragoons and 17th Regiment of (Light) Dragoons were sent to America during the war. These units almost always operated in small detachments (at Cowpens in 1781, there were only 50 troopers from the 17th Light Dragoons present).
Most "British" cavalry in the war were actually Loyalist Americans. During the war, there were over a dozen Loyalist cavalry units formed to fight for the Crown. To whit: Banastre Tarleton infamous green-coated British Legion (a mixed force of infantry and light cavalry), John Smcoe's Queen's Rangers (also a mixed force of infantry and light cavalry), Diemar's Hussars (escaped German prisoners from Saratoga who rejoined the British cause and were attached to other Loyalist units), the partly-black King's American Dragoons, and many more. It was simpler and cheaper to form cavalry in America, rather than bringing over cavalry from England, even if the quality of Loyalist cavalry was rather ... variable.
To get back to your question: why didn't the British use more horsemen?
1: Past experience: The British hadn't used much cavalry in the French and Indian War, and they'd won that conflict. America was also regarded by some British officers as too rough and rugged for massed cavalry actions. Hundreds of mounted men couldn't easily wheel and charge through heavy timber.
2: Fodder Horses eat and drink a lot. Even a few hundred horses needed tons of hay every day. For a logistically-strained British army, that created a problem. Donald Moran writes:
A major problem that affected the use of Cavalry, both British and American, was forage. It was scarce, difficult to transport, and prevented them from massing their cavalry. Recognizing this, Washington spent a great deal of effort denying the British access to forage. On territory they controlled, such as Long Island, he had raids launched across Long Island Sound, whose purpose was the destruction of stored fodder. Sergeant Elijah Churchill of Sheldon’s 2nd Dragoons led one such raid and was awarded one of three "Badges of Military Merit", the equivalent of today’s Medal of Honor. He and his troopers destroyed 300 tons of hay.
During the "Forage War" in New Jersey between January and March 1777, Washington's Continentals and militiamen raided and harassed Anglo-Hessian forces. Destroying, capturing, or otherwise denying them fodder for their forces was a key objective for the Patriot forces.
Lack of fodder was a major limiting factor for both sides. Invasion of Canada in 1775 failed partly because of a lack of fodder. Patriot General Philip Schuyler was forced to delay his invasion until late summer, when there was enough fodder for his cavalry mounts and pack horses. After some initial successes, Schuyler late start meant he was trapped in Canada when winter set in. Without enough of a foothold in Canada to supply their men, and unable to supply their forces from the colonies, the American campaign in Canada failed.
Throughout the war, American and British commanders had to constantly grapple with the fodder dilemma. Keeping cavalry with the rest if the army meant they'd quickly denude the surrounding area of fodder. Sending the cavalry out to forage solved this problem, but left them vulnerable to attack and out of place when they might be needed. Fodder was also expensive, since farmers had to be compensated for the use of their hay and pastures.
British forces had similar supply constraints, as Todd W. Braisted writes:
In New York in December 1777, with but a small garrison and no cavalry to speak of, 907 horses [(used to draw wagons and artillery)] were drawing rations daily. With each horse drawing about fourteen pounds of hay (or forage) daily, that added up to over six tons of hay every day. With the army's return from Philadelphia, and around a thousand cavalry horses, that number skyrocketed. A return made out on August 30 by Deputy Commissary George Brinley showed there was 748 tons of hay on hand, an additional 1,750 tons yet to be harvested within the lines and more within reach in Westchester. 'A very Considerable quantity of Salt Hay might be had at Long Island, and Staten Island, provided the Farmers could be assured of a price that would pay them for cutting it...
3: Transport Transporting horses hundreds of miles across the Atlantic was a brutal endeavour. Hundreds of horses died in the crossing. The ones that did make it often needed weeks to build up their strength after weeks of stressful inactivity.
Even short sea journeys could be horse-killers, as Donald Moran writes:
Lieutenant General Sir Henry Clinton arrived with 10,000 additional troops, besieged Charleston [in 1780] and forced General Lincoln to surrender. ... [T]he voyage from New York was a rough passage and many of the horses died en route, and the surviving horses were unfit for duty.
So, shipping thousands of cavalrymen and dragoons and tens of thousands of horses to America would have been very, very costly in expensive horseflesh.
4: Cost Training a cavalrymen, buying his mounts (exhaustion, disease, and battle killed cavalry horses at an alarming rate), and provisioning him and his horse was an expensive endeavour. In the relatively small battle of Waxhaws in 1780, Banastre Tarleton lost 31 horses in a single charge. Prices varied, but a cavalry horse of this period could cost 5-20 times the monthly salary of a cavalry trooper. Raising and sustaining infantry units was thus considerably cheaper.
5: Little institutional interest in cavalry: In 1775, the British Army only had three regiments of Household Cavalry [heavy, unarmored cavalry], seven regiments of Dragoon Guards [not actually "Guards," really just medium cavalry], and six regiments of Light Dragoons [light cavalry/mounted infantry]. Britain's army lagged far behind the cavalry maestros of Europe: the French, Hungarians, and the Poles (indeed, one Polish volunteer, Casimir Pulaski, would become the "Father of American Cavalry" and help the Continental Army build up its cavalry branch).
6: Training So why not raise new cavalry units in Britain? Why not raise Loyalist cavalry units? Well, to some extent the British did so the latter. Loyalist troopers in Simcoe's Rangers and Tarelton's Legion enjoyed some considerable successes during the war.
However, raising new cavalry formations was far easier said than done. A cavalry recruit had to learn the fundamentals of his new trade: the use of saber, pistol, and carbine, the art of mounting and dismonting, controlling a horse with a mind of its own, and spending hours caring for the needs of his horse. The horses themselves had to be broken and trained to respond to their riders' commands. And once man and rider were ready, there was more to do. Officers and men had to manage an unwieldy mass of hundreds of horses and riders. They had to learn to move and fight as one, controlled by little more than bugle calls and a few shouted orders. All in all, it could take two to three years to form a decent cavalry reigment from scratch. All the while, precious men, horses, and resources were being used up for no immediate gain. Infantry were needed in America, and they were needed right away, so they were the British army's priority.
7: Few Patriot cavalry Many of the same problems that affected the British affected the Patriot side even more. In the early part of the war Patriot cavalrymen didn't even have enough sabers and had to rely on crude swords or captured sabers! By June 1777, there were only 260 American troopers in Washington's army. Although their numbers would grow, the Continental Army and Patriot militias never had more than a few thousands mounted troops in the entire country!
With such limited American cavalry, the British had little incentive to make a counter-move by splurging to bringing in more cavalry of their own.
Sources:
"The Birth of the American Cavalry" by Donald N. Moran
The Logistics of War Ed. Beth Scott, et al.
Grand Forage 1778: The Battleground Around New York City by Todd W. Braisted