I hear what you're saying, but I still think you need to be careful when drawing parallels between what TFL (an arm of local government) and Heathrow (a private company) can and can't do.
If it helps, a number of other non-local-authority English airports (Gatwick, EMA, Bristol, for example) all issue invoices in the form of PCNs to follow up non-payment of their drop-off charges. I can't see any reason why Heathrow should be able to justify being a special case.
But yes, time will tell.