In England in the 1600/1700s, would most people have been proficient at riding horses? Did only elites know how to ride?

You ask a reasonable question because it is unclear to many moderns that horses were actually of vastly different quality and suited to very different types of work during the sixteenth-eighteenth centuries.

To answer your broader enquiry, it seems that horse riding was common to see in England, but the numbers of people who rode in comparison the population was small. Although as early as 1558 the Venetian Ambassador claimed that “there is no male or female peasant who does not ride on horseback, and miserable must that man be who follows his cart on foot. Thus the rustic on horseback drives the oxen or horses of his team”. [CSPV, 6.3 (1557-1558), p.1672] This is clearly an exaggeration, or possibly a misuse of the term “peasant”. Surviving probate inventories suggest only the wealthier land-owning farmers owned saddle horses, most riding horse in private ownership were limited to the gentry and nobility who had the funds to support them.

Most people could not afford a horse as maintaining and keeping a horse was rather expensive, even a poor-quality one unsuitable for riding. While horses were indeed everywhere and commonplace, being the preeminent mode of transport and carriage on land, they were not in any measure majority riding horses. Most horses owned by others were not specifically saddle-horses but were instead working horses whose role changed as needed by their owner. The gait of the horse was a big factor in how comfortable, and useful it was to ride, which dictated its job. Military horses most often had a trotting gait, hunters were ‘rackers’ or ‘pacers’ whose gait suited ‘best speed’ over rough country, while horses who were ‘pads’ were relied on for their amble gait for long, paced journeys. Horses with a trotting gait were least preferred for long journeys, as the speed and bounce of the gait made it damn uncomfortable to ride on long journeys in a normal saddle. Military saddles of the time were higher, more supported, more expensive, and best suited to carry a man on a trotting horse (at the trot was usually the precursor to the charge).

Also, while people could indeed ride cart-horses or other work horses, doing so too much meant the horse could risk developing a different gait. Gervase Markham in his Cheape and Good Husbandry: For the Well-ordering of All Beasts and Fowls, I (1614), p.5 advised “Above all things observe never to put your draught beasts to the saddle, for that alters their pace, and hurts them in their labour” adversely affecting their ability to work a cart.

To ride a horse any distance actually requires a considerable skill, one mastered and honed with practice. If you did not actually posses a saddle horse, then you had to rent one. If you didn’t own a horse, and this was often the case for the poorer in the countryside and for most people in towns, you could rent one from a Hackneyman, who could range from a private individual who rented out their horse, to the postmaster and innkeepers who kept small stables. In both cases the rental resulted from existing conditions – the owner already owned a horse he did not use all the time so made profit by renting it, and postmasters and innkeepers already had the facilities to stable and keep horses overnight as part of their role, so keeping a supply on-hand for rental seemed logical. This still was not a cheap option. In Elizabeth I’s time renting a horse was between 2 and 3 pence per mile (far beyond the measure of a poorer individual).

TLDR: Horses were commonplace, riding them less so except at need for the majority of horse owners other than the wealthy. The majority of the population did not ride. More can be said about breeds, types, and native horse development during this period, but that is fundamentally your answer. I am happy to answer any follow up question you might have.