This is because of the extreme subsidies air traffic receives. If it were not for these rail would be very competitive.
I don't think so. Railway lines are the epitome of lost fund subsidies by the state. You need to build and maintain a new line for each direct connection you want to create. The construction cost of a railway line, let alone a line that supports high-speed rail, eclipses the cost of a brand new airport.
Once a railway line is in place, getting a train to travel from point A to B costs peanuts.
See this Greenpeace study (in German) about how much cheaper flying is than train rides and why:
https://www.tagesschau.de/inland/regional/nordrheinwestfalen....
So exempting tax (equivalently not charging for externalities) is indeed a subsidy, a transfer of wealth from government (and so society) to the airline operators.