The problem with rail is the capacity issues affect the entire line. If one station has busy platforms and the rest are all quiet, you can't put additional trains on that line (unless it's a terminal station and there's a turning circle before that station and you are willing to run trains that don't go there). Also if one line shares fast and slow trains, that kills capacity.
One airport can share fast and slow planes, and capacity problems at one airport don't affect other airports.
I guess I'm writing mostly from the British perspective. My comment about fuel taxes paying for the roads is (or was) true for the UK.
Since privatisation at the start of the 1990s decades of decline in ridership turned around and now more people use the railways than ever. In the UK there are lines everywhere that are not only at capacity but drastically over-capacity, e.g. there's a train departing from Euston on a Friday night that carries more than 2x the number of people it was designed for. Despite that Crossrail - one of the biggest civil engineering projects in Europe - isn't even open yet, the country just committed to an even more eyewateringly expensive new rail line that won't be open for over a decade. Primarily justified on capacity grounds.
Heathrow expansion is also very expensive of course. But there's really only one airport in the entire country that has such severe capacity issues, and the expansion is difficult primarily because it's surrounded on all sides by urban development like motorways and villages that have to be demolished and relocated. In the case of the motorway, it's also carrying such large capacity that it can't actually be closed for extended periods, so it has to be moved whilst "live".
I think the USA has a different set of issues because the railways are used mostly for cargo rather than passengers. The large distances involved means air travel is the only feasible mode of transport for many routes, and Americans take internal flights all the time whereas for most Brits I know, internal travel is almost always car or train. Business people take internal flights, and even then only quite rarely. I've traveled Acela and indeed it also wasn't heavily used.