It's straight up economics.  If a coin's face value is worth less than its melt value, there's a good chance a lot of those coins are going to make their way to metal recyclers.  That isn't the case nowadays, but it was many years ago.

And who cares if the statement in the title is true?  That's about as useful as saying "in the long run, we're all dead".  I don't even agree with it, because there are still coins and jewelry that have survived hundreds of years or more.  That's a long enough run as a coin or necklace or whatever for me.  Plus these days even junk silver coins don't get melted.  If the value of gold or silver plummets, you might see some melting, but I think those days are long gone.