It seems to me that if there was truly a shortage of pennies, banks could offer to pay 2 cents for every penny someone turned in (still far cheaper than minting a new one) and enough people would pull out their penny jars and cash them in.

Now we've unlocked a new lucrative business model where we can cut the customer out entirely! Simply buy pennies from the bank on the dollar and sell them back for 100% profit...

That...actually seems economically sound, but it's also a strong argument for the idea that pennies are effectively worthless.