Commercial insurance covers a combination of driver and car. You can get insurance for someone to drive a car that they don't own as part of a contract (e.g. with Uber) that they did not directly sign or benefit from.

All you have to do is be upfront about it, and willing to pay.

Yes, it does cost more to cover employees as opposed to owner/operators of a business venture, but it it is not impossible (if it were taxis would not be able to exist).

What is probably true is the default insurance given out by Uber will not cover people not listed on the policy, and only the person contracted with Uber is listed on the Uber-backed policy.