AFAIK non-profits can receive donations bound to a specific purpose - they just don't like the accounting headache?

I wonder why the current structure is set up like that. I mean I don't think it's by accident – if it were set up like that by accident, surely, it could have been set up so people could direct donations directly at Firefox, if the CEO wanted that.

The structure is set up that because it reflects the goals of Mozilla. Firefox the browser is not the end goal, it's a means to making the internet a better place. And that's what the Foundation is about. The corporation also pays dividends to the Foundation in order to help fund their efforts, because it's the Foundation's work that matters more (in the long term) than Firefox.

It was established precisely so that the Corporation could pursue commercial operations that the Foundation legally couldn't before that, like business partnerships.

Also Mozilla Foundation didn't qualify as a public charity since it received less than 1/3rd of its revenue via donations, though I don't know if it can be considered as one now (15 years after the split).