How about the broken picks are linked to the Mining Hat (like the Big Fish Hat, only for mining)?
Like you can either drop them (25 weight) or NPC them (0z), or take them back to the miner who tells you "Thanks for bringing back these picks, it really saves me a bit of trouble getting the materials. Keep it up and I promise I'll get you something nice!"
And after 10,000 broken picks, presto, Mining Hat!