I think you have a misunderstanding of what's copyrightable.

> This idea that they own the very idea of 'boy wizard who goes to school' is insane and plays right into this flawed and pernicious idea.

Copyright protects an expression of an idea, but not the idea itself. No one can legally stop you from writing your own story about a boy wizard who goes to school.

Nintendo fan games can be released (even sold) if they changed it up a bit so they're no longer associated with that IP.

> Copyright is important but does/should not extend to every time I want to print out a picture of a boy wizard for my kid.

You can. No one will sue you. No one will send you a cease and desist. It happens when you try to print Harry Potter and try to make a business out of it.

> No one can legally stop you from writing your own story about a boy wizard who goes to school

This is doing exactly that, except for an image. They didn't ask for a picture of Harry Potter, they asked for an image of a boy wizard and were prevented. The actual IP holder probably didn't implement these specific restrictions but it falls into the category I'm talking about, the idea that they own everything. It shouldn't be up to the image generator to say I can't get a picture of Harry Potter, it should be up to me whether or not I try to sell that image and risk getting busted for it. That's where copyright comes into play, as far I understand.