Americans love their cars. I may want my car to express my individuality. Or, no matter how convenient and efficient it might be to ride the same SkyTran car thirty people have used before me today, I may think like one friend who said, "I don't want to ride in some smelly public pod".
Others have practical reasons to specially outfit a SkyTran car. A tradesman may want to replace the second seat with a larger set of tools than he can carry. Someone with a handicap may need special adaptations (assuming that she can access the portals at her chosen destinations.) SkyTranFreightDelivery companies will need vehicles designed for automatic loading. EmergencyServices ambulances and police vehicles need special design to accomodate stretchers, K-9 dog cages and handcuffed suspects.
Assuming that the solution to WhoOwnsSkyTran allows for privately-owned and outfitted vehicles, there are several technical and economic issues.
In special cases like EmergencyServices, the answer is simple: the portal will be pre-empted. As the paramedics and/or ambulance pod approach, audio and video signals will warn customers to leave both platforms (cooperation will no doubt be legally as well as morally required, just as moving aside for an ambulance is today). The waiting pods close their doors and leave empty, and the portal is closed for ordinary business until the emergency is taken care of.
Private vehicles for less urgent purposes would require more subtle methods. In an unmodified ("serial") portal, vehicles for the general public and owners of private vehicles would need to be properly interleaved. Owners will need some kind of wireless communicator to summon their vehicle, but there also needs to be a way to ensure that vehicles are available for any customers ahead of them before it reaches the portal; so it will circle in a holding pattern and arrive only when its owner is near the portal. When she reaches the boarding location, any waiting standard pods are dispatched before her vehicle (and replaced with others that come in right after it).
Supporting private vehicles and other specialized vehicles (like a handicap-accessible pod for a wheelchair, or SkyTranFreightDelivery, bring up the possibility of a different, "parallel" portal. Instead of a single, "serial" stream of vehicles, the track diverges one or more times to parallel platforms with special purposes; so one kind of traffic doesn't need to wait for the other. The proposed $10,000 cost of an entire basic portal implies that forking the track between the main acceleration and deceleration sections should only cost a few thousand. This will certainly happen in special locations that justify it. The challenge will be accomodating individualized vehicles in the majority of standard locations.
Another angle is that private vehicles provide a way to differentiate customers at different levels of the demand curve. A private system can maximize its revenue; a public one can subsidise low-income or handicapped people with this extra revenue.