It probably could increase quite a lot because more people might be willing to live far away and enjoy cheaper / more housing if it means that commuting won't be dead time. With self driving cars you could actually work during your commute. As you point out, a number of existing commuters might also transfer from train to car travel.

You're forgetting the unoccupied trips. There will be unoccupied trips running errands even at rush hour, because why not, you're not sitting in traffic, just your car.

Also you seem to be thinking more of the robotaxis model, but the article is talking about the more likely future of everyone continuing to own their own car, only now it can drive. If cars are driving back empty after a commute then rush hour will have roughly twice the traffic.

One way to prevent congestion would be to instead of paying for the distance traveled, you simply paid for the time the car is on the road, giving a financial incentive to send the car outside of rush hours. Would probably also cause people to use alternative routes rather than everyone trying to use the shortest one when is congested.

Yes, this is one of the main solutions proposed by the linked article.

Wouldn't that be a trip that someone would normally be driving anyway? Also wouldn't it make sense to have a store deliver to lots of people on one circuit, this reducing traffic?

No, once you don't need to drive it personally you can run a lot more errands, and thus will. See the linked article.

And look at how much more random stuff people buy when a click of a mouse gets it delivered to your house vs having to go to stores looking for it.

If one car is going in a loop and delivering to lots of people it would be less time on the road than all the cars of the individual errands.

This doesn't seem to be the world we're headed towards though. People like owning their own personal vehicles and aren't likely to give that up.

What are you basing that on? In dense cities with good transportation infrastructure people don't own cars nearly as much and the people that do own cars use them less.

I think you are confusing necessity with desire.