Does a proportion of the infrastructure costs that are required to get people to and from the airport, be it the Piccadilly Line, Heathrow Express, motorway, road junctions etc not come from the general tax payer?
Actually, it's often the other way round. Planning permission is often granted to developers but subject to a Section 106 clause which commits the developer to pay for road access, improved infrastructure, social housing, community centres, doctors surgeries and the like.
No doubt the beneficiaries of R3 (BAA) would be expected to incorporate into their scheme a raft of social/community benefits from which they make no direct gain. Section 106 is about to be changed, but the principle will remain that developers are expected to pay a social premium.