You can keep your minivan. Or you just hire a vehicle that has the room you need only when you need it.
Uber already is pretty fast. I just used it in upstate New York north of Albany and there was hardly any wait. Any wait that exists will go down once robotic vehicles becomes dominant.
And pricing will also be important to make that happen. People who want less than 5 minutes wait can pay more, but that will be pretty reasonable priced. It's cheaper to pay for a vehicle to wait for 5 minutes or even 20 than it ever could be to make it wait for 3 hours or 12 hours.
And that IS essentially what we do when we have a vehicle on constant standby. We are paying for the fixed costs of having it wait around all the time. There's a daily cost to tie up that capital. It's 1/30 of your monthly payment. And an hourly cost.
In any case, individuals will be able to choose the option that suits them. And go back and forth. Some days you may value immediate pickup and you could rental a vehicle by the day. Other days you may not care and want to save money. Or you use your vehicle to care around tons of stuff and it's better for you to own. It's all good.