I don't know Islam's exact role in Iranian government, but it clearly has one, as it determined the last revolution there. But more importantly, they're in a region of monarchies and dictatorships. Not saying they can't be a democracy, it's just a big assumption to make that they would've been, like I really don't see any reason.

Btw, lots of Arabic influence in Iranian culture. Yeah separate Indoeuropean language, but Arabic alphabet, tons of Arabic loanwords, and required Arabic lessons partially to help read the Quran. There's a reason the Pahlavis wanted to emphasize Aryan pride instead.