It's a reasonable hypothesis, but the problem is that you have two 'squishy' things - reason and democracy - and you're asking for a coherent answer. It's not impossible, but it's a difficult one.
As you know, the word democracy long predates the enlightenment, there are some examples here. They include not just obvious democracies like Athens, but more obscure ones like early Native American societies. Many of these societies had slavery, property and gender-restricted franchises, but then so did most nineteenth-century democracies. I can just about credit that the ancient Greeks had an enlightenment-style attitude towards reason, but extending that to Mesopotamia, India and the pre-Columbian Americas seems a bit much. Not saying that they didn't have reason or a belief in it, just that if they did it must be a universal human quality whether you live in a democracy or an autocracy.
The main difference between ancient democracies and modern ones is that modern ones cover an entire country. Large countries existed in antiquity, but only as autocracies. What made it possible to run a democracy where the demos is spread over thousands of miles, and can never know one other, let alone meet in one place to make decisions?
The answer: the printing press. That allowed people to talk to the whole country at once, and for everybody to be informed in a timely way about what the country's problems were, and what was proposed to do about them. The printing press also cemented a belief in human reason. A medieval bishop could believe - or pretend to - that reason was some special quality which only the best people (aristocrats) could possess. The expensiveness and consequent scarcity of books, and concomitant widespread illiteracy, lent some plausibility to those claims. Medieval European paper and ink was more scarce and dear than that used in ancient Rome, although it did last longer.
In other words, both belief in reason and modern democracy are an outcome of the same innovation: printing.