It will never be Uber flicked a switch and replaced all their human drivers. It will be a very gradual transition over to self driving that may never reach more than 25-30% of all trips.
Is that worth the billions of dollars they're pouring into self driving? https://www.uber.com/newsroom/company-info/ says they complete 14 million trips a day. That is 5.1 billion trips a year (and growing). 10% of that is a 500 million trips a year. Assuming the average trip is 13$, that would be $6.5 billion per year in revenue, I think that's more than enough to turn a significant profit even accounting for R&D, Capital expenses to maintain the fleet etc.
The reason self driving will be a reality is purely economic. Noone's developing it to change the world or reduce the number of people dying in accidents or whatever shpiel they cook up.