So which system would society rather have to allocate a limited resource? First come first serve? Person who can pay more? Physical fights? A lottery?

There is no single answer, but the more essential an item is the less accepting society is with "Person who can pay more" approach. Eggs are one of the more fundamental ingredients in the diets of many Americans.

Increased pricing signals the issue to consumers who would otherwise be unaware though, that's why it generally works. I don't know the technical definition of price gouging, but I typically associate it with something that consumers NEED, i.e. Hurricane comes through and wipes out 20% of homes, hotels raise prices when people are literally stranded == gouging. Raising eggs prices... wouldn't people just not buy / buy less eggs? (That's what we chose). Seems like that would also keep supply from bottoming out as quickly.

Yep, price gouging is just regular market-based pricing but when it occurs during an emergency. Why some people think it mysteriously stops being a good system during times of emergencies is beyond me.

>Why some people think it mysteriously stops being a good system during times of emergencies is beyond me.

It's pretty simple: this 'self-regulation' is most harmful to the people who are most vulnerable during emergencies. The rich always have a lot more options to deal with scarcity.

But if that's true, isn't it also true in times of non emergencies? This just seems like an argument against market pricing in general.

That is the dirty little secret of price gouging laws. It is just the system acting as normal so labeling that behavior as immoral is basically an indictment of the entire economic system.