Your quote is high due to need for lots of bonding coat required to match height / thickness of lower sections which remain. It’s hard work, requires skill, and takes time – has to be applied in layers. Depending on thickness you can easily lose a bag of bonding coat per square metre so possibly up £350 or more in plaster.
Your life would be easier if you had no original plaster on the walls. Then you could dot and dab the wall with aster oars and simply skim. That job would probably cost around £700 – £800 of which maybe materials £150 (multi finish is around £11 a bag at the mo).
Your current situation is about as difficult / expensive as it gets without getting into special aste types etc.