The cost of the transport element financed by the taxpayer is small compared to the cost of the project itself. LHR is not getting something for nothing. The company is taking on massive risk financing a large scale project that will benefit the UK and bring in extra money for years to come.
Who will finance the alternative? Who will finance the access to Boris Island?
Lets be clear about the timescales here. If the project was given the go ahead tomorrow it would likely be 2030 before it became operational. The M25/M4 around Heathrow will see significant investment in that time.
Also lets remember that a huge chunk of that taxpayer funded portion and indeed the LHR financed element comes straight back to the taxpayer in the form of VAT and other taxes on the construction cost alone.