this ID card would give every private pilots license holder (who passes the checks) unrestricted and unescorted access to the Heathrow apron as well
I'll disregard for a moment the fact that the subject of this thread is total fantasy (for now anyway), to describe the working of the airport IDs that I'm familiar with.
In Spain, AENA-run airports have an ID system which gives access to different areas of a single airport.
Essentially, firemen and the airport manager have unrestricted access to all areas, while everyone else is restricted to one or more specific areas only, e.g., administrative staff do not have airside access, and airside workers do not have automatic access to the office areas.
Each airport has its own ID, so depending on what you do, you could have an ID from more than one airport, or more than one ID for a specific airport.
Pilots are never required to have an airport ID. What is needed is a pilot's licence, some form of official ID, and if going airside, proof of a good reason for being there--this usually takes the form of either a stamped copy of your flight plan, or you just tell them your aircraft's registration number (which they cross-check with what is parked on the apron). This is more to do with safety than security btw, as apparently it's relatively common for PPLs and others to go planespotting airside
(this according to a Guardia Civil I was chatting with once).
However, should you wish to have an airport ID it is certainly possible to get one as a based pilot. All you need is a letter from your employer or flight school, etc., and usually it only takes about ten minutes. It has two big advantages: free parking and discount at the airport cafeteria (or access to the even cheaper airport canteen where available). The ID itself is of course free of all cost.
So there you go, FWIW. I am of course not endorsing Orwellian-style IDs such as the stuff authorities in the UK have wet dreams about. In fact I am not endorsing IDs at all, although at the same time I have nothing against the system run by AENA as described above. At least it's straightforward and you get some perks.