The compressor clutch is electrically operated, the pulley wheel is driven all the time by the same belt.
Did the workshop that did the AC service say if the clutch was not operating or if it is is it slipping ?
When the clutch engages it can be heard and seen if you can locate the compressor under the bonnet.
Don't confuse the clutch noise with the AC cooling fan starting up.
Knowing the price of some AC compressor jobs at around £1k , £470 sounds reasonable. Where you are there must be recommended AC workshops around to get alternative advice.
Further to the above, today Thursday,while getting some new front tyres fitted at my son's workshop, he tells me that the price of AC compressors has come down recently so it may pay to look around.
Vic.