It seems to be a deliberate ploy by the big supermarkets (excluding Aldi) to mix in house depots with contractors running others.
Sainsbury’s Haydock in house
Sainsbury’s Stoke DHL
Sainsbury Sherburn Wincanton
Co-op Castlewood in house
Co-op Avonmouth GXO
Co-op Coventry Stobarts
I’ve found some contractors get the same store discount as in house drivers, while others don’t, as with everything it varies from contract to contract.