BHX certainly does not need a London tag on it. If one thing in life is guaranteed its the fact that it will never happen.
Planning application will be approved without the need for a Public enquiry. So they don't need the help.
You are right though, in that BHX could advertise the new rail times to its advantage and steal some of that London Traffic. How long is the journey from Heathrow/Stansted/Gatwick to
central London?