It would make much more sense for it to be unified with the MTA subway, with more connectivity. My dream would be a single seat ride from Newark Airport to Atlantic Terminal in Brooklyn, via Newark Penn Station and World Trade Center.
There was also once a proposal to have the 7 train go out to Seacaucus Junction, which would be huge for accessing the East Side for NJers.
Taxi companies don't like rail going to US airports. With JFK you have to take one train, then change, then another train, then you're at the equivalent of Paddington. Would be like having to change at Hayes+Harlington. Luton has this, but that's the odd one.
From Newark I think it's a bus to a train station, then a train to NY Penn.
The "default" way to get from Heathrow/Gatwick/Stansted to London is a train from the terminal to the centre, typically multiple options.
The default way in the US is taxi. Massively inefficient, but a lot of people make a lot of money from it. Why have a plane load of people on a single train costing $5k, when you can have 200 taxis costing $20k, generating 4 times the GDP.
In London, you also have the Piccadilly line depending upon where you're going and how much luggage you have (many of the London tube stations are pretty awful in terms of accessibility as I discovered on a trip where I had heavier luggage than normal last year).
Since 2022 there's also the Elizabeth Line, which uses the same tracks as the Heathrow Express, but then continues under Central London and far to the east.
It takes 27 rather than 15 minutes to Paddington, but it's also half the price of the Heathrow Express.
Right. But I usually stay near Trafalgar Square and the Piccadilly line is much more convenient for that. (Though I will be taking the Elizabeth line from LHR my next trip because I'm initially going to Shoreditch. And, yes, Heathrow Express is something of a rip-off which the airport steers you towards.)
If you're talking about underwater, I don't think that's the problem compared to running track through populated areas, whether that's through tunnels, cut-and-cover, at-grade or viaduct.