Democratication came at the hands of improved communications, improved education, the industrial revolution and the agricultural revolution. Not at the hands of cheap weapons.
The invention of the printing press democratised knowledge. Books became significantly cheaper to make and were more widely distributed. It was possible for a wider range of thoughts to be made known. Pamphlets on political subjects were easily printed and widely read. Other advances in communication have increased further this democratisation of information. This makes it easier to mobilise the masses to action. Example, the English Civil War saw rampant propaganda via pamphleteers. The weaponry on both sides was quite similar, both having mounted cavalry, muskets and pikemen.
The Enlightenment in Europe brought improved eduction. People understood complex political systems and had the knowledge to use them to their advantage. Knowledge can be seen as continuing tool of democratisation throughout the developing world. It can be see that as the population's education improves, kings, tyrants and dictators fall and are replaced by more representative systems.
The industrial revolution changed the finances of the world. Countries no longer derived their wealth from the land. Other sources of money meant that other people, than the landed gentry, would have the financial clout to affect the politics of a country.
Agricultural Revolutions changed the balance of power too. People shifted off the land into the cities as the farms were consolidated and mechanised. A feudal system requires a peasantry and after the Agricultural Revolution in the UK, there was little peasantry left. Power shifted to the new cities and to the workers.
A second argument can be made that cheap weapons are available throughout the world today. However, rather than democratise the world, they have been tools of dictatorship and tyranny. In the less developed areas of Africa, South America, and Asia, where the previously stated factors had not yet taken effect, we saw a great many dictators. Throughout the 20th Century, dictators in the third world have used cheap weapons, such as the AK47, to build there empire and stifle democracy.