According to the linked Wikipedia article, he did not go broke from the gold rush. He went broke because he invested the pickaxe windfall in land, and when his wife divorced him, the judge ruled he had to pay her 50%, but since he was 100% in land he had to sell it. (The article is not clear why he couldn't deed her 50% of it, or only sell 50%. Maybe it happened during a bad market, he had a deadline, etc.)
So maybe if the AI pickaxe sellers get divorced it could lead to poor financial results, but I'm not sure his story is applicable otherwise.
Basically every tech company likes to say they are selling pickaxes, but basically no VC funded company matches that model. To actually come out ahead selling pickaxes you had to pocket a profit on each one you sold.
If you sell your pickaxes at a loss to gain market share, or pour all of your revenue into rapid pickaxe store expansion, you’re going to be just as broke as prospectors when the boom goes bust.
In a gold rush, the folks that sell pickaxes make a reliable living.