The Monarchy, like all rulers, is able to rule because they have the backing of powerful factions. No matter how powerful you think a ruler, ultimately they have to delegate a lot of control to those they want to carry out their orders. There is no difference in any country. Even in the US, were the DoD and some other departments able to successfully coordinate, they could easily take over the government, at least for a short while. That they don't is more a nature of how problematic it would be to do anything useful after that was accomplished, and that it would likely be short lived.
My minimal understanding of the Magna Carta, as taught to an American in middle school, is that the Nobility forced the monarchy to sign a document ensuring they are subject to certain laws like others are (but not necessarily all the laws the average citizen is subject to), which in essence increased the power of the Nobility. They were able to do this because most the military might of the time was in the hands of the nobles. The Monarchy might have had a larger army than any single noble, but the combined military strength of the nobility dwarfed the Monarchy (wars were generally fought by calling on the nobility to supply soldiers).
My point was though that the loss of powers of the Monarchy in the UK had little to do with the citizens, it was a power grab by the nobility. And then I assume eventually much of the strength of the nobility was ceded to parliament and then parliament was eventually opened to commoners (I'm getting into speculation here, but speculation based on snippets of history I do know so I think is likely not far off the mark).
In the end, the ability of the citizens to have guns specifically had very little to do with it I think. AIUI, there are other countries with mostly figurehead monarchies which I suspect had a more gradual shift in power as well, and wasn't explicitly at the threat of violence. It may have been spurred by the sight of that happening in other countries, but really that's more a realization that the populace is learning that they have power in quantity, guns or not, and those ruling them needed to contend with that reality.
Attributing it all to guns is a vast oversimplification in my eyes. It may have had far more to do with the printing press.