By that logic, why is anything a public good? If I drink water from a public water fountain why is that in the public's interest?
If there were no fountain, then many people would get water bottles and that's too big of a hassle.
Defining a public good is quite hard, and any definition will most likely other controversial terms like "reasonably" or "foreseeable" in them. However, I think a public good is something that can be shared by many people in reasonable amounts, and that cannot reasonably be attained by the average individual. It is usually characterized by its scale.
So for example, you can't build your own park, but you can use the public park without blasting your music on max volume.
Right, and if there was no public transit, then many more people would need to buy and drive cars, and that's not great for the public's interest either.
I think there is a difference between providing the opportunity and giving things out for free.
[1] Both because there are higher safety expectations for railways than for roads and because one of the inherent working principles of railways is to bundle individual demand into larger units (a.k.a. trains) that can be moved across the network en bloc. If everybody tried to operate their own individual little train across the network the way you do with driving your car, things wouldn't work out.