Counterpoint:
In every war that involves a substantial conventional component, there has been a huge ammunition shortage. Even though people know this, the phenomenon still exists.
Private industry isn't going to maintain huge "burst" capacity. It's simply completely uneconomical.
It's possible that a sort of public-private partnership could exist where the government pays companies to maintain huge excess capacity, but I don't really see a substantial difference between that and the government just owning everything.
And it's important that this excess capacity gets used often, otherwise there's a big risk that it won't work when it's eventually called upon.
Which isn't to say that there's no place for private enterprise to supply ammunition to the government. Just that it probably shouldn't be the only source.