My experience (in Portugal) is that they provide a sort of safety net and offer small but precious favors, thanks to being owned by a Person (and not a faceless corporation) who you can get to know and establish a trust relationship.

Our corner stores provide items on credit for local residents when money stretches thin before the paycheck arrives, they deliver for free to the poor old lady who can't leave her home, or they hold your home keys if you needed to give them to someone when you were away. In exchange, we buy some stuff from them, even if it's a bit more expensive.

Plus, some have relationships with local farmers and other small scale producers, and so can get some genuinely better products than supermarkets. I always bought national fruits and bread from them.