This is because funds allocated to projects in London and the SE cannot be allocated twice
Potentially they can - though admittedly not at the same time. Take, for instance Manchester's £300 million housing investment fund, part of their recent devolution deal. It works on the basis of an initial investment unlocking a development site; properties get constructed and sold releasing value which is then ploughed back into the fund and reinvested in the next scheme. It's called a revolving fund and is quite common.
You could regard investment in infrastructure in the same way. Passengers travelling from Heathrow generate in excess of £1 billion of APD a year for the Treasury. Investing say £5bn in infrastructure to support expansion at Heathrow will generate increased APD so the 'investment' could in effect be repaid within maybe 10 years?
Interestingly (as this is the Liverpool thread, after all) Liverpool City Council is to consider a recommendation next Friday to provide investment finance for Liverpool Airport's short to medium term expansion, primarily because they see it as an asset to the City Region, but also because it should generate a return on that investment over time.
Comparing the price-tag for delivering a long-haul capable runway at two different locations in the UK could hardly be more relevant. London prices are higher than regional prices but doesn't 100x more expensive in the same country ring any alarm bells with you?
When you finally start comparing like with like Shed, people may take your argument more seriously, but your insistence on comparing R2 at Manchester(which was indeed a new runway and associated taxiways) with the third runway at Heathrow (which includes a new runway, taxiway, aprons, terminal, satellites, road and rail infrastructure and over £1bn of compensation to local residents) undermines your case.