In the interest of being factual I have to point out that it's not quite that bad. You're quoting the export price, but the US Military gets them for closer to $1M per because they funded the development of the system and have ongoing contracts for delivery.

Still not great when you're spending a million a shot to intercept $20,000 dollar drones.

The bigger issue though is production capacity. Raytheon can't just magically quadruple their output overnight. Patriot is an extremely complex device that requires skilled personnel and complex tooling to manufacture, as well as a LOT of safety procedures (it is a missile after all), in addition to the rare earth requirements you noted. Even if the US offers to start buying them at export price Raytheon can't just make that production capacity appear out of nowhere. And in order for them to scale up capacity they're going to want guarantees of ongoing purchases so that they don't just do a bunch of capex to meet a short term need.

This is why maybe you don't want a publicly traded military industrial complex beholden only to shareholders; you end up with an expensive system that doesn't win wars and a systematically impoverished country that needs war to sustain itself.