While technically true perhaps, in reality it isn't unless you're talking about John Travolta.
It isn't just technically true, and it applies to more people than just John Travolta. (Why you use him as an example is a mystery.) It is true to the extent that if you are flying someone in a rental aircraft you cannot let them pay for the whole rental. That is considered to be "compensation" because you get to log the PIC time and can credit time towards currency requirements. THAT's how serious the "no compensation" part of the FAA regs are.
Private pilots fly very small (and comparatively inexpensive) aircraft, like 50-year-old Cessnas, which can't hold more than 4 people
Private pilots also fly multi-engine aircraft that were built in the last five years. If I wanted to, and had the money, I could drop a few million (at least 5.25 according to this, and 9-11 pax) on a King Air and fly it as a private pilot. The private pilot's license does not limit the pilot to "50-year-old Cessnas", and making that kind of statement shows you are ignorant of the reality of general aviation. The fact that you seem to think that "commercial" flight is limited to "commercial airlines" is also a clear sign of ignorance. Those 50-year-old Cessnas find service hauling passengers around for pay, too, even if there is a limited number of seats. One of my very first flights was on a Cessna (I think it was a 172) as a passenger going from Kansas City to Manhattan Kansas. The local FBO will fly people anywhere they want to go in the 172 they operate.
It simply isn't possible, in practice, for a private pilot to have the lives of hundreds of people in his hands.
"Number of passengers" has nothing to do with the difference between "private pilot" and "commercial pilot", and it is just as rare for a commercial pilot to have hundreds of passengers, too. By the time you get hired to fly the big iron you need to be an ATP -- one step above commercial. As a private pilot I can fly 14 of my dearest friends around in a Cessna Caravan (the beautiful aircraft that FedEx uses for cargo), but I couldn't charge any of them, nor can I charge any of the one passengers I could ferry around in a Skycatcher. No, the differentiation between needing a "private" versus "commercial" is not the number of passengers allowed, it is the paid vs. non-paid status.
As for your point #2, I don't see how it's any different if I hop in my friend's or neighbor's or relative's car as a passenger. I don't actually know that they've been maintaining their car properly.
You know them, and that will give you some indication of whether they are casual or rigorous about their standards of maintenance. You also know where they live, so if something bad happens they will be relatively easy to track down and they won't likely flee the area.
Also, it's pretty unusual for cabs to have more than 2 passengers,
So? Killing just 2 in a poorly maintained car is ok, three is not?
This isn't like planes at all.
The point of an analogy is not to show a congruence between two things, it is to highlight the similarities. Commercial aviation is similar to commercial ground transport because of the lack of ability to know the risks. The fact that the ground transport rarely gets above 0 AGL and aviation almost always does is irrelevant, as is the alleged number of passengers (which as I've already said, has nothing to do with the differentiation between "private" and "commercial" pilots.) I know of no limitation on number of passengers in Part 91 of the FAR, and I expect there is only an indirect limitation on a private pilot flying a 747 in that it would be hard to get the type rating for, or find someone who would rent one out for a joyride to, a PP AMEL. If someone does know a relevant FAR, it would be interesting to hear it.
So my whole point here is: if we're not going to bother having ANY standards on the roads at all, for our drivers or our cars, then why should cabs?
Your conditional clause fails, the "then" clause is not invoked. Were there no standards at all, then your point would have some merit. It would still be valid to have standards for commercial operations when there are no standards for private action, just as the sport pilot license has much less training required, but also has less privileges. And as the proposed removal of the third class medical for PP privileges would not imply there should be no medical certification for commercial operations.