Hi muscat and welcome to FT

It does sound a bit of a rigmarole you have to go through. Is there not a lift you could use with your luggage as opposed to removing poles and getting on escalators? (Sorry to sound a bit dumb, I've not yet been to GVA but that will soon be rectified). If not I would think the best option would be, as you suggest, to drop the passengers off first and then drop off the car.

Unfortunately in these times of essential cost cutting, I think it unlikely that an additional staff member could be utilised as you'd like. And much as BA might like the airport to change their procedures, it would be very difficult for them to do it for one airline and not all the rest.