I think you are missing the point VK. Those of us living in the SWT and Southern areas would have to travel to London Waterloo or Victoria when we don’t have to now, despite there being no direct rail link, there is the coach via Woking to LHR.
You are right about the aims of Crossrail but it was not designed to serve an airport in Kent. Don’t under estimate the power of the bird lobby, they only have to visit the place at Christmas and it becomes protected. The birds that is, not the lobby!