Second, I doubt that the private gun owners of the United States consume as much 5.56/.223 as the entirety of NATO. The cartridge is widely available, popular, and comparatively inexpensive because of the large military demand for it.

I'd be further very surprised if the combined US private consumption of all other centerfire[0] rifle cartridges adds up to the private US consumption of 5.56. The economies of scale behind 5.56 are huge. 5.56 can be had for under $1/round in bulk. Most other centerfire rifle cartridges aren't commonly sold in bulk (.308/7.62x51 being the only other one that comes to mind, which is also a NATO round). Basically anything that you might reasonably hunt deer[0] with is around $2/round.

The takeaway from all this should be that if NATO went to war, we might have to worry about the supply of 5.56. While there's a very visible part of the US gun culture that's dominated by the popularity of 5.56 and AR-pattern weapons, I don't think we can "thank" it for the abundance of 5.56 relative to the shortage of everything else that's being used in Ukraine.

[0] The .22 S/L/LR variants and the .17 derivatives are rimfire, which are entirely different from a manufacturing standpoint.

[1] I'm not going to get into the debate of whether 5.56 is suitable as a deer cartridge. I don't know. 5.56 is in broadly the same class of cartridges as the classic deer cartridges, at least from a manufacturing standpoint.