I live in the UK and caught a train to London then a Eurotunnel train to Paris and it was probably quicker and easier when you account for time spent in departures and going through airport security.
It seems that the cost of an additional passenger on a train is almost zero, so why do trains not attempt to reduce per-ticket prices to increase capacity and become more competitive against low cost airlines?
In your example, the train from London to Paris in my experience has almost always been nearly triple the cost of a flight.
It is. But the cost of the first passenger is massive.
Think of all these rails (and the terrain under it you have to pay a hefty price for), all the bridges etc.
You mention the Paris-London train, just the tunnel did cost $21 billions, add the high speed rails before and after it and you are probably at close to $40 billions in infrastructure before the first train can do the trip.
Trains are orders of magnitude greener than planes, but unless you massively tax planes, or massively subsidize trains, they'll always be more expensive than planes.