So why was Imperial Iran so critical?
You may want to rethink your timespan. It’s really short. :)
The time span I mentioned is over 40 years-sorry I wasn’t clear enough-or till the Russian revolution,when the Russians stopped focussing(temporarily) on Iran.Iran was critical because of it’s location-pretty much similar as that of Afghanistan,though it had the advantage of a large number of ports as well as land borders with the Persian Gulf states-essentially a bridge between the Persian Gulf & the Indian colonies.
Not to mention oil,after 1900 or so.