If spellsilver is the kind of thing it sounds like, where you can filter it out using a series of chemical reactions, then spellsilver has not always existed since the beginning of time. It got turned from non-spellsilver into spellsilver at some point.
Unfortunately 'at some point' is billions of years ago inside the center of exploding stars, which is beyond what even Keltham knows cheap ways to do at scale.
But the point is, spellsilver exists; most things that exist can be made out of other things that are not themselves. The question is, what is the cheapest way of getting more of a material? And this, for spellsilver, is almost surely mining it and purifying it with acids. That doesn't mean Keltham isn't going to check the other routes.
If what distinguishes spellsilver from depleted spellsilver is a tiny weight difference between two metals that otherwise seem chemically to be exactly equivalent, this means Keltham cannot realistically figure out how to discharge depleted spellsilver. If it's not that, he should go on thinking. They don't know, so he'll go on thinking.
If there were, say, a ten-thousand-pound mass of pure spellsilver lying around, would there be any good way to detect that from a hundred thousand miles away?