Why would I own my own car in this scenario (vs paying one of the autonomous driving companies to send a car my way when I need one)?

Because if you own your car you leave your golf clubs in there just in case you get invited for a round.

If you are the type who is willing to be seen in a used car you can save a lot of money since the rental car needs to be newer cars just in case someone who wouldn't be seen in a used car wants one - and this adds a lot of cost.