One thing that some cities have done where awkward infrastructure is required to get a train to the airport is to, essentially, borrow money to do it, and make the fares to the airport very high to compensate.

Notably, getting to Brussels airport, which takes about 15 minutes from Brussels Nord, costs about 15 euro. For a 15 minute train journey. Hands-down the most expensive train per minute (or per km) I've ever been on. But, at least in theory, it's paying for this thing: https://en.wikipedia.org/wiki/Diabolo_project

(That's by no means the only one; lots of airports are in awkward places so running rail to them is expensive, and it's common for it to be paid for by special, more expensive services. And people use them.)