You deal with peak demand by having a supply that is equal to the peak. Automatic Rideshare companies can self balance over a large area, watch the predictable trends that show up during a year and so on and will print money and thus have a lot of capital.
You're also forgetting that most existing cars on the road have way more seating capacity than needed, since they need to account for the maximum number of people who will use that car at once. A family of four needs a car with four seats, even if most of the time that car will only be carrying one person to work and back throughout the week. Self driving cars will be much more efficient in that respect.
That assumes that nearby places have different peak times, which is probably a flawed assumption.
By offering discounted fares at off-peak times, and charging premium fares at the busiest times, you can smooth out the peaks while optimising profitability.