You'd be surprised at how much fits in a good backpack or set of panniers. A bike trailer is also an option if you really need a lot, and as a side benefit you can carry your kids in it when you're not grocery shopping. Delivery is another reasonable option that probably causes less pollution and traffic than everybody driving their own car to the store.
You'd be surprised at how often your bike is gone when you walk out of the grocery store. If city governments want to encourage cycling then they'll have to take bike theft as seriously as auto theft and stop coddling thieves.
Your bike is not getting clipped at vons in broad daylight while you are inside for 30 mins and my vons borders many a bike chop businesses operating out of tents on the sidewalk. Overnight? Absolutely.
