The video is not necessarily wrong. The short answer is, it's a bit more complicated than that, but since the second amendment is a source of never-ending heated debate, it is slightly more difficult to see the forest through the trees, as they say. There were a couple of main reasons for the second amendment. One was a check on the federal power, as you mentioned, the other was pragmatism, or a concern for future preparedness. Since it's a little more speculative, we'll start with the latter. As you know, a significant portion of the continental army were "minutemen". Not all minutemen came ready armed with there own musket, and the continental army was already having problems equipping their properly trained soldiers. Most were expected to provide their own, many of which were not fit for service, and 15% without a firearm entirely. So, when it came time for the constitutional convention, this was a concern for both federalist and anti-federalists alike. Now, there is substantial evidence to suggest OP's main point (of a concern for standing armies) was a very large driving force behind the drafting of the second amendment. In fact, in an original draft of the bill of rights proposed by the anti-federalists there is actually a clause following the right to bear arms that prohibits standing armies. (Cornell, Saul "The Other Founders", The University of North Carolina Press, 1999) However, regarding the runaway slave theory, it's not impossible, but doubtful. See, when the constitution was drafted in 1787, cotton was expensive to produce and the slave trade was generally perceived at the time to be on its way out. When the cotton gin was invented in 1794, cotton became a genuine cash crop, and the slave trade saw a comeback. In conclusion, there was definitely a concern for standing armies on part of the anti-federalists, and with a mutual concern for ill-prepared militia, it was more or less a win-win for everybody in the battle for ratification. Although it should be noted that these fact in particular get extremely skewed when it comes to biased second amendment articles. I am making my best effort to present them in a neutral fashion. And the slave patrol, I can't say it could not have been proposed at some point, but I've yet to see/find any good sources, and given the climate surrounding the slave trade at the time, I'm saying unlikely.

The Second Amendment was largely designed to guarantee the existence of local militias capable of both 1) providing defense against localized insurrection and frontier attacks and 2) checking the power of standing armies and abuses.

My own knowledge concerns the latter; I'd appreciate it if anyone can discuss how much supporters of the Bill of Rights were actually concerned with the militia's ability to fight rebels and Indians.

One of the more controversial aspects of the Constitution was Article 1, Section 8, which gave Congress the unrestricted power to raise armies. Anti-Federalists expressed grave concerns about the implications of a peacetime standing armies. See, for instance, The tenth letter of Brutus.

This fear of standing armies is a result of centuries of English monarchs using their armies to persecute domestic enemies and infringe upon the rights to liberty and property. This began with William the Conqueror, who used his Norman army for domestic policing of local barons that opposed his rule. These fears were exacerbated during the reign of Charles I, who sought to greatly expand royal prerogative.The Interregnum did little to quell these misgivings. Cromwell's New Model Army was by far the most expensive army in English history, and its refusal of Parliament’s 1647 order to disband stoked additional fears that the army sought to seize political power. These fears proved accurate the following year, when the Rump used the military to purge Parliament following its passaged of the Militia Ordinance, which, in an effort to counteract the military’s power, placed local militias under Parliamentary command.

The Militia Ordinance reveals two important themes in Anglo-American history. Firstly, legislators and citizens were deeply concerned about executive authority over standing armies. Secondly, citizen militias were viewed as a potential check on such armies.

Although the Restoration temporarily quieted criticism of standing armies, the issue resurfaced in the Glorious Revolution. The 1689 English Bill of Rights, presented to William II, condemned his predecessor James II for “raising and keeping a standing army within this kingdom in time of peace without consent of Parliament,” boldly declaring such conduct to be “against law.” The same year, Parliament passed the Mutiny Act (very similar to the power on Article 1, Section 8), creating a permanent standing army under the notable caveat, inspired by deep distrust of the military, that it be reauthorized by an annual legislative vote.

English colonists across the Atlantic espoused similar concerns about standing armies as a result of their own history. Governor Edmund Andros of the tyrannical Dominion of New England used soldiers to collect taxes and impose martial law in the late 1680s. The unpopular 1765 Quartering Act required colonists to fund barracks and supply lodging and led to soldier-civilian tensions that culminated in the 1770 Boston Massacre. The 1774 Quartering Act, part of the so-called “Intolerable Acts,” was even more egregious to the colonists in mandating that they provide soldiers with rooms in private homes.

In contrast to standing armies, militias were extremely popular in seventeenth-century England. James Harrington's famous work The Commonwealth of Oceana promoted a Roman-inspired civilian militia that, through constant readiness, would promote civic virtue and dissuade invasion of Britain. Members of the 1659 Long Parliament went further in their extolling of the militia, citing its ability to act as a counterbalance to the New Model Army.

Americans inherited this alacrity for the militia, which would prove to be a valuable tool in the protection of English colonies, most notably in the French and Indian War, the North American theater of the worldwide Seven Years’ War. It held great cultural and social significance as well. Local parades and training brought communities together and served as a right of passage for young men. Militias checked royal authority in the colonies, participating in both Bacon’s Rebellion and a revolt against the despotic reign of Governor Edmund Andros in the Dominion of New England. The prestige of militias was greatly enhanced by the American Revolution. Minutemen launched and won the war’s initial clashes and vital early battles in the Hudson Valley region that enabled Washington’s ragtag army to survive. The militia would continue to be an instrumental resource to the Continental Army throughout the war and would later be romanticized and praised by American writers across the political spectrum. An anti-Federalist’s article in the Pennsylvania Packet, citing the militia’s victories at Lexington, Bunker Hill, and Burgoyne, asked whether “a well-regulated militia [were] sufficient for every purpose of internal defense? And which of you, my fellow citizens is afraid of any invasion […] that our brave militia would not be able immediately to repel?” Hamilton in Federalist 25 observed the wartime militias have “by their valor on numerous occasions, erected eternal monuments to their fame.”

So what's all this have to do with the Second Amendment? The Federalist Papers (23-29) present a powerful argument in favor of a standing army (largely relying on the constitutional requirement that it be reauthorized by Congress every two years). They also note that local militias are inadequate to counter major threats and rebellions. This frightened anti-Federalists who cited the English and American experience with the militia as an important counterweight to standing armies. They feared that the nationalization of the militias would enable tyranny and oppression. Hence the Second Amendment, which, by guaranteeing the right to bear arms in order to form a well-regulated militia, ensures that states and local citizens could check national power.

Sources: Shawn Humphrey and Bradley A. Hansen, “Constraining the State’s Ability to Employ Force: The Standing Army Debates, 1697-99,” Journal of Institutional Economics 6, no. 2 (June 2010)
John Phillip Reid, In Defiance of the Law: The Standing Army Controversy, The Two Constitutions, and the Coming of the American Revolution (Chapel Hill: The University of North Carolina Press, 1981).
William S. Fields and David T. Hardy, “The Third Amendment and the Issue of the Maintenance of Standing Armies: A Legal History,” The American Journal of Legal History 35, no. 4 (October 1991).
John D. Waghelstein, “Regulars, Irregulars, and Militia: The American Revolution,” Small Wars & Insurgencies 6, no. 2 (1995).
Lois G. Schwoerer, “No Standing Armies!” The Antiarmy Ideology in Seventeenth-Century England (Baltimore: The Johns Hopkins University Press, 1974).

Edit: grammar and formatting