SS,
First you try to apply motorway budgeting to a new airport, now you suggest the same engineering could apply to noise reduction. Let's stick with the engineering challenge this airport poses in its own right, and solve with devices which are relevant to aviation.
BHX has huge grass berms which provide some shielding to houses between the airport terminal complex and Martson Green.
Although I don't agree with this airport, it should be fairly obvious that it provides a massive net reduction in noise footprint, compared to Heathrow.
However, noise reduction alone does not create a business case for a new airport, nor is it remotely fesible to suggest welfare budgets can be slashed to pay for it - then you are just moving a problem of individual 'scroungers' to corporate welfaree 'scrounging' - of course the problem isn't s simple as that either, but that is one for Jetblast.
So, I repeat - £50bn investment, how do you propose an ROI on commercial terms, given the risks?