I have other more pressing things to worry about than if someone occasionally touches bottles while refilling them. They might, yet, touch the bottle to my glass, horrors. It may not be ideal, but that large bottle is not being used to fill very many glasses or bottles, and in practice BA staff tend not to touch the containers when they pour.
I ask them to fill up my own water bottle quite often. I'm trying to use less single-use plastic in the world.
I'm just waiting for the cries of horror if your water bottle is refilled from the potable water tank.