Few people just like to hang out on trains, although I do remember the one guy on Reddit who did all his coursework while cruising around Switzerland and then got trapped in a railyard. However, plenty of people will choose a long commute or to visit a more distant destination if it's cheap enough. Extreme example, but I once knew a person that drove a full 2 hours each way to work. Through a far more densely populated area.

I don't and couldn't really have empirical evidence that people would overuse free public transit, but the I think the theory is strong. Generally, people will travel less if travel is more expensive. If travel is provided at cost instead, they'll avoid it unless the value to them of the travel exceeds the cost to other people to provide it and bear the side effects.

Another thought: Flights In some places there's a government air carrier, and they fulfill the same basic function of getting people from point A to point B. Usually they're not considered public transport, but then you have cases like the small arctic communities in my country, which are filled with very poor people and can only be accessed by plane. Should we make an exception? That's where things might get complicated.