The US had a huge amount of help from France overthrowing the British. Including full support of the French navy and large amounts of French professional soldiers. An inconvenient fact that Americans try hard to forget.
So that refines the question to: how important was the private ownership of guns in the American victory at Saratoga? Well, the Wikipedia article referenced above says that, "militiamen and supplies continued to pour into the American camp, including critical increases in ammunition, which had been severely depleted in the first battle."
"General Fraser was mortally wounded in this phase of the battle, . . . The fall of Fraser and the arrival of Ten Broeck's large militia brigade (which roughly equaled the entire British reconnaissance force in size), broke the British will, and they began a disorganized retreat toward their entrenchments."
So that got me curious about what "militia" meant exactly during this time frame. Well, General Ten Broeck's Wikipedia page says that 2 years prior to the action described in the last paragraph General Ten Broeck was colonel of the Albany County militia, which has a Wikipedia page, which starts as follows: "The Albany County militia was the colonial militia of Albany County, New York. Drawn from the general male population, by law all male inhabitants from 15 to 55 had to be enrolled in militia companies."
>To place any dependence on the Militia, is, assuredly, resting upon a broken staff. Men just dragged from the tender Scenes of domestic life; unaccustomed to the din of Arms; totally unacquainted with every kind of military skill, which being followed by a want of confidence in themselves, when opposed to Troops regularly trained, disciplined, and appointed, superior in knowledge and superior in Arms, makes them timid, and ready to fly from their own shadows ... if I was called upon to declare upon Oath, whether the Militia have been most serviceable or hurtful upon the whole, I should subscribe to the latter.
