Not having to own a car to travel by car will of course increase driving, or being driven really. A situation where there is only self-driving cars is probably decades away at he minimum. And that doesn't really solve inherent throughput or speed issues.
Even if self-driving cars are 50% of the fleet, these cars drive better and can collectively work around traffic jams, so even human-driven cars benefit from automation.
Your visible marginal cost (i.e. mostly your fuel) may be low but the vast bulk of car costs are proportional to mileage--especially in areas where salt in various forms (winter roads, marine) isn't an issue.
It doesn't really matter if it's lower than the fixed costs, it matters whether it's lower than these hypothetical rented cars, and most likely, it is.
