Quite apart from the payment issues / mess, this whole thread raises the question "what to you put on a public transport map or general promotion?". And whilst we criticise, no-one has come up with anything that suits everyone. The presence of a dozen official tube maps -
https://tfl.gov.uk/maps/track/tube - shows the scale of the problem; I'm sure
TfL» would prefer a single map!
Logic suggests to me a complete public transport map of all turn up and ride systems. But that's going to get overcomplex when you pile on the longer and shorter distance stuff, buses and trains. And does all include the tube to Olympia, the train from Paddington to High Wycombe, or the bus that diverts past St Winifred's on school days? And what about that demand responsive bus, and night buses? Such a map - but interactive (zoomable, and with time of day / week elements that grey out services not available near a chosen time) .... [as I write, I'm remembering all the massive issues ....]
Putting myself in the traveller's / visitor's shoes, I do like the TfL area specific bus maps (Paddington example below). And that's against my own initial view. They're simple, a quick read, show you where to get on in an immediate area and where direct services go. The addition of underground / Rail services would be useful, and I wonder about an indication of very common connections onwards (the map for Bromley North showing train connections at Grove Park for Charing Cross, for example).
For Heathrow ... perhaps it exists ... such a map as "your onward transport options" ... one for the central area, and one each for terminals 4 and 5.
Map referred to in text - click on image to link to original map as a .pdf