They had one but unfortunately for the Iranians the British and the Americans got rid of him and put the puppet Shah in his place.
While by all accounts Mossadegh was a great man and the one who nationalized our oil, I don't think we can say with 100% certainty he'd be a democratic leader, seeing he only was in charge for about a year or so. Also, I don't agree Shah was a puppet. Yes, in 1953, he owed his rule to the coup and he had great relations with the West, but he wasn't a Western puppet for the entirety of his rule (He pointed out Jewish over-representation in US Media and Banks for example, said if you remove Khomeini's turbon, it has a "Made in UK" sign tattooed to his head" etc. Shah did a lot to modernize Iranian society through his White Revolution and actually genuinely loved and cared for Iran, but he made some big strategic mistakes like trying to change too much too fast, not killing off Khomeini early on in 1964 instead of sending him to exile, not addressing inequality adequately and abolition of political parties to except for one.
The fact is, people now dream of the Shah era and see it in a nostalgic way. While Shah wasn't as popular and effective as his father was, but he was still significantly and by millions of miles better than the current mob who don't even call the country "IRAN", but rather "Islamic Republic".