Ultimately this is a geometry problem. Cars are by far the least space-efficient method of transporting people; eventually your roads just can't accommodate any more traffic. If there's enough demand to visit a given area then anything that doesn't minimize cars will just make things worse.
And parking is a time sink. There's a place in my city that has huge parking garages with lots of parking but you still have to drive through a few levels of the garage, park, and then walk back down a few flights of stairs, then walk to your destination. I just park my bike right outside of my destination with the wheel lock. Street parking is always awful in populated cities, and I never have to worry about it. I always park right at my destination, where ever it is.
In suburbia, cars are faster because the average distance per trip is a lot longer. But it's ironic that the reason why the average distance is longer is BECAUSE it was built for cars so everything gets spread out! Cars are a solution to a problem that they created.
Not true in practice even if true in theory; in many places the average full-size bus contains fewer people than would fit in a minivan.
The problem is you need 5-10 years of reliable public transit in an area before non-transit users begin to convert (or transit users begin to move in).
The bus might have less than 5 people on it at any given moment you observe, but over its >2 hour route it transports dozens or hundreds of people between stops.
A full minivan ranks surprisingly high on all of them, which is one of the reasons it's disappointing that carpooling and such aren't discussed as much anymore.
I really wish someone would be a modern city from scratch.
(sidenote: I'm pretty sure it started because few guys with minivans noticed that some areas were under-connected by buses and just started a business driving those routes back-and-forth on a schedule. Then some regulations were slapped on top. So it looks like public transport have tendency to spontaneously appear in denser areas)
In Mexico it was always a decision of "do I walk a bit further to get the jitney and pay a bit more, but likely go faster (it stops picking up passengers when full and just goes until the first drop-off), or do I get on a bus sooner for cheaper but likely slower?"