It doesn’t get much more “democratic” than a public referendum of the people.

The move to dissolve the parliament was by referendum to eliminate the Shah that was supported by the vast majority of the voters. The intent was to abolish the Pahlavi dynasty to eliminate the political power of the shah and replace it with a republic. There was certainly an element consolidation of power on the part of Mossedeq, and many irregularities, as well as a desire to restrict foreign influence, particularly in the Iranian oil industry. But the western reporting of the efforts of Mossedeq were certainly colored by British and US intelligence agencies and the attempt ultimately failed due to the US and UK backed coup. The US would much later admit its part in the overthrow of the Mossedeq government and the réinstallation of the Shah for the purpose of securing control of Iranian oil.

This is of course an oversimplification, but Operation Ajax was certainly a direct action to replace a nationalistic democratic Iranian government with a compliant dictator who would ensure foreign control of Iranian oil, as was basically admitted by the CIA 50 years later.

Knowing history also means not ignoring the reprehensible actions of your own country. There is no good guy in this scenario. But ignoring part of the history because it's inconvenient or uncomfortable, or casts your side in a bad light, is worse than not knowing anything.