If there is that much demand for flights out of LHR then I fail to see why the taxpayer should pay for any of the expansion. I would allow LHR (and LGW) to expand as much as they want but only if it is privately financed - after all why shouldn't those getting the benefits of expansion (the passengers) pay for those benefits.
Same question for nuclear power stations or the Channel Tunnel. They're not traditional businesses, they're former UK assets flogged off and run privately whilst remaining key pieces of national infrastructure. So you either partner with the operator to tie them into other parts of UK PLC, i.e. Crossrail or TFL, or do it wholly private. The Heathrow Express was partly built on that basis and as a result was (and still is?) the most expensive railway per mile on the planet, because it didn't benefit from the same subsidy as the rest of the network. Wholly private enterprise at this level is impractical BUT I agree a greater degree of risk should be placed with the private operator than some of the more "Private reward, public risk" fiascos of late. But if you want anything built that's so intertwined into wider public spending, it can't, and shouldn't be done without HMG getting involved.