Your assumption is that people buy based on price only. Far from it - they still have to physically go to the store to buy it and so many other factors then come into play. Are they familiar with the store? Does it have free parking? Is the service good? What is their return policy?
I agree with your main point that small stores might not like such a system but then to assume that all buyers will make a decision based solely on price is flawed.
If you're buying a $100 item (let's say a DVD player that a friend recommended) and you've always bought that item from a store previously because back when you first bought a DVD player you shopped around and found this shop had the best service and prices, it's not worth shopping around again because you're probably only going to save $5 or $10 and going to another 5 stores (which will take an hour because they're all over the town). You go to your favourite store because past experience dictates it's probably the best place.
What if you're standing outside the store and decide to check out a service that will price check for you across EVERY store; you type in the item and it says "this item is $90 at store y" and you notice store y is just 2 doors away. What do you do, do you shop at your regular store or do you shop at the new one? The majority of people will go with the new one, it's stupid not to walk an extra 10 feet to save $10, the new store might even be better!
I agree with your main point that small stores might not like such a system but then to assume that all buyers will make a decision based solely on price is flawed.