Small shot was absolutely widely available:
“At first, single shot was used for target-shooting and four-footed game; by the 1540s multiple shot was also used, at this date cut from lead sheet. By the end of his life Henry possessed 41 such “Haile Shotte peics”, clearly used for birds. These were obviously pretty deadly, as an Act of 1548 at-tempted to ban the “shoting of hayle-shot wherby an infinite sort of fowle is killed and much gaym therby distroyed” and lamented their uselessness to military training. These weapons, loaded with shot, can be regarded as the remote ancestors of the modern shotgun.”
https://www.thefield.co.uk/shooting/history-shotgun-shooting-26590
“In 1686, Richard Blome’s “The Gentleman’s Recreation” became the first English publication portraying the shooting of flying birds. By the early 18th century, overhead shots were becoming a fairly common practice.
In 1776, the first recorded use of the term “shotgun” arose in a Kentucky publication called “Frontier Language of the West” by James Fenimore Cooper. This term separated the smoothbore shotgun from the rifled musket as advancements in technology greatly changed firearm barrel designs.”
https://blog.gritrsports.com/the-history-of-the-shotgun/
Jay