Uber and iphone

I always owned an Android phone. An iPhone is not worth the money or hype.

Recording user interaction has been a part of many apps it is not new , you can check uxcam and other websites which does a similar thing.

The amount of data any analytics framework collects is unimaginable and many of popular apps in the market use more than one for sure afaik. The best part of using iPhone is you can limit tracking by enabling limit ad tracking ( settings > privacy > advertising)

App permissions are a problem that needs addressing but this isn't a permission, it's a hidden from the public direct invasion of privacy.
They have allowed a company known for data mining and tracking its customers unlimited access to everything on your screen at any time without informing the users their phones are compromised.
Have a think about what's on your phone, 'nice' pictures, bank log in details, secure or sensitive documents? All would be available to Uber and you had no idea they could see it.
Moving on, how about government employees of whatever sort, Uber could see every word they ever typed and every document they ever read.
It's a spy's wet dream.

By the way, the standard permissions allow them to access your camera - careful where your phone is when you get out of the shower.