The problem is that all the supermarkets copy each other.
If Sainsburys were selling milk for 10p a pint so would Tescos so as not to lose custom. If Sainsburys charge £5.00 for a loaf, so would Tescos, Tescos would charge £4.99 maximising profits.
Agreed that they do offer competition to the larger chains like BP, and can have a +ve on prices, but I think the main concern is that when crude oil goes up, pump prices respond immediately, when crude goes down, pump prices don't for a long time.