The way I heard it, trains in Japan aren't (state) subsidized, but rather they are real estate businesses where "subsidized" cheap traffic into and between train stations drives increased real estate value and thus commercial rents.

If you think about it, the same could be said of state subsidized public transport, where increased economic activity due to improved traffic (getting people to/from jobs, shops and their homes) can increase tax revenue which can then be spent on public transport subsidies, turning them revenue positive. Of course whether most state subsidized systems actually live up to those aspirations is a bit more questionable.

Do public roads work without heavy subsidies?

Something can just be a public good without delivering profit.

No form of transport works without subsidies, except maybe walking.