They did anyway, in practice.
In contrast to what some have written here, democratic ideals which came rather close to those of Classical Antiquity were highly popular and widely practiced during the High and Late Middle Ages.
Just not on the level of state, kingdom or empire. The democratic polis of the European Middle Ages was the village of the Alpine space, Southern and Western German-speaking HRE, bits of France and Spain, Transylvania and Scandinavia.
The village ruled itself in general assemblies; it elected mayors for a limited period of time; it managed its own finances; it elected sheriffs; laymen juries dealt with criminals. The male heads of households were the citizens, much like in ancient poleis.
Much of this was achieved in the 13th century already. It became threatened and was undermined in most places by centralising states which followed a different logic. Against this de-democratisation, millions of peasants rose in dozens of rebellions.
If you want our Modern Age more democratic, at least in the sense of Antiquity and the Middle Ages (i.e. excluding servants, women etc.), you need these rebellions to succeed and establish stronger, larger confederacies.
Ancient Greek democratic theory would be a nice add-on, something some leaders might use in grand speeches and scholars might write their theses on in their ivory towers. But it wouldn`t make much of a difference because the Late Middle Ages had both their democratic praxis and a theory to base it on - the Christian theory that all humans are images of God.
A spontaneous idea as to how a Medieval-Modernity transition might have gone on in a more democratic fashion:
- Make the Swiss Coniuratio and models of Christian Republics like Calvinist Geneva more active in foreign policy, forging alliances with free towns, supporting peasant rebellions, so that Swiss-like confederacies are established e.g. in Bavaria, Tyrol. From there, you`d have to have the league of democratic confederacies struggle really hard against the forming proto-absolutist monarchies, though.
Actually, I´d be quite interesting to see how such a timeline would develop... Unfortunately, I don`t have time to write it myself.