Once the silver was melted down it would lose some of its value as a median of exchange, because it would no longer be of a specific weight and would be less recognizable, which is part of the benefit of coining silver and gold. It would still retain its value in silver, but you would have to find someone that was willing to trade their good/service for a silver spoon. Moreover, the value of a medium of exchange is it’s standard measure and uniformity. By melting the coins the weight changes, so every person that would get the spoon in exchange would have to verify its weight/ consistency and make the economic calculation with regard to their specific product.
As illustration of the reputation/recognition issue, Rothbard in his treatise “What has the Government Done to Our Money” reveals that the term Dollar is not in fact American in its origin at all but was derived from the Count of Schlick who lived in Joachim’s Valley. It was originally termed a “thaler”. It gained prominence because it had a good reputation for their uniformity. Currencies that are debased lose that reputation (in addition to the actual precious metal lost from the coin) and people will seek a better median of exchange (cows, salt, cigarettes) if one is available or if legal tender laws prohibit competing currencies, Gresham’s Law will pay out and only the debased coins will circulate.
Unfortunately, modern U.S. currency has no connection to precious metals. Precious metals continue to retain their value because they are scarce, meaning it takes time and effort to extract them and create useful products from them. In contrast U.S. Money can be created for little to nothing, and are not open to competition because of those legal tender laws. The government can increase the supply of money out of thin air, but they can’t mine, extract, and coin precious metals in the same manner, which added a level of stability and security.
Hope this helps. Rothbard’s book is short, easy to read and is much more cogent then I. I’m sure Tom and Dr. Herberner could have more to add, but thats the gist.