When the supply of something is lower than the number of people that are price sensitive and desperate, capping prices will still leave desperate people without what they want. Many mechanisms that appear to be more fair, like free lotteries, end up distorting things in other ways: If it's a free lottery, you bet people will sign up without being desperate.

One might have an aesthetically favorite version of deciding who is going to go without, but there's always going to be a problem with just that. What really helps is increasing the supply of what people want. And for that, it's hard to beat increasing rewards to the suppliers.

Don't high price signals in an efficient market serve the community by increasing supply? It's hard to argue that there's a better way to do it that doesn't inadvertently decrease the supply and instead become a lottery

That is an incoherent, meaningless sentence. You think people should be allowed to pay high prices, but no one should be allowed to accept their money? What do you think it means to be allowed to pay for something?

Why would you cater to those willing to pay any price? They’re a tiny minority of people.