The argument that we should stop minting them because they cost more to produce than their face value falls flat for me.

A penny is not a single use item. The cost of production must be depreciated across the thousands of transactions in which it is used and then compared to the economic benefit of its existence.

It may be true that the economic benefit of a penny is less than the production cost but I don’t see anyone making that case.

What will it actually cost the US economy to stop minting these coins?

How long will they remain in circulation until they are no longer accepted for payment?