UAE wealth and influence comes from oil, there is no fudging around that fact. And they do live in the 12th century in some respects, they're just doing it within the comfort of their 21st century homes, with their 21st century cars, weapons, and infrastructure. The UAE is a dictatorship with strong extremist religious convictions and a medieval social values system that can only be described as appalling.

These things don't go away simply because they're our "allies" or because criticizing extremist religious practices is frowned upon in the West.

You are probably mixing up UAE and Dubai. The UAE consists of 7 emirates, including the capital Abu Dhabi and Dubai. The GDP of Dubai is dependent on services, trade and a bit of industry, and they have little oil. Abu Dhabi is oil-rich. A big chunk of its GDP, and the net GDP of the UAE in general, is oil.