AIUI the highway has up to 3 components:
carriageway, usually called road
footway, usually called pavement (I consider pavement to be paved with slabs or blocks; asphalt isn't paving) or footpath (footpath can be public (the highway isn't public, i.e. no right of way), permitted (open to public by permission of the owner) or private)
verge
Double yellow lines apply to the highway, so parking on the footway is still an offence but no action taken as it doesn't inconvenience motons.
You're conflating lots of things...
Footway = path restricted to pedestrians alongside a highway.
Footpath = path restricted to pedestrians away from a highway.
All parts of highways (carriageway, footways and verges)
are public and, unless specifically restricted (e.g. motorways and certain A roads), pedestrians, horse riders, cyclists, and non-mechanically propelled vehicles
do have right of way. Mechanically propelled vehicles are on highways 'by permission' which is why they have to be registered, insured, and their drivers licensed.