That's actually a great thought experiment. We see it all the time with cars and how it fails.
Right now we don't have pods, we have cars. Cars run everywhere. So in your rush hour example, we gridlock with cars getting to work. Now they're all needing to be placed somewhere for the workday so we need all this parking. Then they have to leave to go home. Again, massive parking. And because all the vehicles are run by humans, there's accidents and slowdowns and death.
Here's how the skytran system works.
They determine peak traffic flow and the number of pods needed to support it. Those are added to the network as necessary. Offline cars are kept back at the depot.
Every stop has sidings for spare pods. You take a pod, the next spare slides up.
So, how do you handle a sports venue? The venue lets the system know the time of the event for scheduling purposes. As fans arrive their vacated pods go back into circulation. No parking lot is needed. What you do have is a heavy number of stops ringing the venue for hundreds of simultaneous offloads. When the event is over, pods have been prepositioning along those sidings. You don't need every pod for every fan then and there, more will be coming as pods depart. But in short order you are emptying the venue and everyone goes home.
Republicans tend to be the biggest detractors of PRT because it cannibalizes the existing customer base for cars with are a) 20th century state-of-the-art PRT and b) tremendously flawed. But it appeals to the American ownership model, I own my car, fuck you I got mine, jack.
The advantage with something like Skytran is it works with a lower passenger density than required for traditioanl mass transit like subways. You could lay a Skytran grid down on top of our low-density sprawlscape.