I am just trying to fashion an intuitive understanding of what the axiom of choice is even saying....

It seems it is a prohibition against too much randomness? it is saying that we disallow collections that are so dissimilar that it would be impossible to formulate a "picking procedure" that applied to all its sets. Is that right?

If that is right, then one can understand it as a constraint of the very idea of a collection.

essentially saying they can be crazy, but not too crazy.

is that right?