The scenario is a cross-country trip in an electric car. What actual, specific advantage does a train or bus offer in this scenario? What problem does it solve better?

It's an electric car, so carbon emissions are low.

Most of the route will be in rural areas in the middle of the night, so the impact on traffic will be minimal.

As for the cost to build and maintain the roads, they are already needed so rural areas are accessible. Wear and tear on roads and bridges isn't much of an issue since heavy vehicles like trucks cause massively disproportionate damage[1]. (A bus might actually be worse than the equivalent number of cars in this respect.)

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[1] See https://blog.ucs.org/dave-cooke/trucks-cause-the-lions-share... . Some studies show that damage varies with the fourth power of axle weight.