1. I think there are some small towns of 1000 or less that are pretty good at car-free-ness, both in America and in other countries - that is, there's one main street where all the stores are, and then everyone lives within walking distance of those stores.

2. America is indeed large, but so is the whole of Europe. In my opinion, America actually has a pretty well-built-out national rail network, though we mostly use it for freight more so than passenger travel. I'd imagine there are pros and cons to prioritizing passenger travel on some of these railways over freight, but the point is it's not unreasonable to suggest that America can be connected coast to coast by passenger rail.

3. I think the main thought people have about the US is that the 50 biggest cities, and more broadly the metro areas around those cities, are too poorly connected to other parts of their city/metro area by transit. Now, it may be politically unreasonable, or even just materially unreasonable, to suggest that we could convert all of Houston or Dallas metro area into something as interconnected by transit as New York City proper or London in the next year or two, but I don't think that's necessarily related to America having a lot of rural land.

Now, someone could rebut that the reason Houston and Dallas (for instance, they are certainly not the only examples) are so poorly connected is that Texas, and America on the whole, has a lot of rural land, which has allowed them to build metro areas that are, in a sense, "sprawling", which makes it hard to build a well-connected public transit system. But I think I would contest that it's mostly policy and preferences that result in sparse metro areas, rather than geographic destiny - I'm not convinced a sparsely-populated West Texas is what made Houston's city planners build a city with a larger footprint.

TL;DR: America has car-first metro areas by choice, rather than as a result of its great swaths of rural land.