> pickers I would imagine would do the opposite (i.e. get rid of the store's oldest stock first).

Maybe if they worked for the store, but why would they do that if they worked for an independent service with ratings and customer support?

Personally, my experience even with first-party shoppers is that they seem to select for good quality. I've never gotten unpleasant produce or near-expired items. I'm sure it happens, but I'm also sure it produces plenty of support calls.