I don't want to comment on if it should be free or not, but public transit and micromobility should always be cheaper than owning+driving a car. Even with NYC's controversial new congestion pricing, it's still cheaper to drive your family to the city than to take them via train.
No it's absolutely not cheaper to drive your family into the city. You said owning+driving a car; did you really take into account ownership costs (maintenance, insurance etc) and driving costs (tolls, parking)? Just the cost of parking alone dwarfs the cost of the train.
Sure but most families need a car anyway, and once you have one the costs of ownership, maintenance, insurance etc. are irrelevant to the costs of individual trips.
Most families in NYC do not need to own a car, and in fact most don't.
But even then we have to consider marginal costs. Owning one car is better than two or three. Here in Texas, it's not uncommon to see households with 4 or 5 cars.
Well 57% is most, although that doesn't account for having multiple kids in one household.
Either way it's close, and the closer you get to Manhattan the higher that number goes. Remember, there are over 8 million people in NYC, and over 12 million during the work day.
You want to buy a second-hand car. What's the first thing you check? Yep, the mileage. The cost of a car is basically down to its depreciation, which is basically down to its mileage.
It's not only the mileage. Age is very important too. Look at any car advert - it's the age that is highlighted first. Mileage only if it is unusually low.
Many components degrade simply due to time, especially things like rubber seals.
Commuting as one person from Mountain View to SF was cheaper with a car including the total ownership cost of the car, nevermind the reasonable discount for "stuff I wand to do with my car other than commute".
And riding caltrain during peak hours for an hour twice a day would violate the Geneva Convention on the treatment of prisoners of war.
Riding Caltrain during peak hours is totally fine and not unpleasant at all, particularly with the new trains (I do it most days and am writing this comment while doing so)
I left California before the new trains were a thing, but peak hours going north starting in Mountain View meant standing room only tightly packed on the world's more unstable train cars.
I took Caltrain regularly pre-pandemic when I had a job in Palo Alto and going to SF. It was not bad at all. Other cities like NYC had way more packed trains.