I would chose Turk rule of Iraq. It was better then Saddam's Iraq or British empire (at least Turk rule of Iraq never had hunger holocausts).
“Which one gave the people a fighting chance at democracy once they gained independence?”
Once you have already gained independence then its up to you to build democracy regardless of who your previous master was. Pakistan was also under British rule, what happened to their democracy. How many countries in the common wealth are “democracies”?
“The Brits weren’t at all perfect, but the Indian politicians (for better and worse) pretty much adopted the Brit system when they got the chance to form their government.”
Worng. Indians adopted their system from the best of many different sources. Indian constitution was adopted from the Irish system.
“Given the large number of countries that have that form of government, I’m thinking that this wasn’t a bad thing.”
You are totally clueless.
“Gengis Khan”
You had your chance at controlling India but your Mongol horde was repelled by the Delhi sultanate.
Europe gained plenty of knowledge from other parts of the world without being made colonies by those more advanced nations (one of which resided in the land of modern day India and Pakistan). And as you've stated in this post, India took its legal system from other countries without those countries ruling over it.