Either humorous or serious answers: can you name a for-profit type of business that would never work if it were nonprofit/not-for-profit? ~

Prostitution: by definition, it's paid.
Those who give freebies for charity go by a different name.

I've never been in business, so really I'm an idiot to attempt an answer,
but this is how I understand it...

Easier to say most businesses wouldn't work unless they made a profit.
That's the stuff left over after every bill is paid, including wages, maintenance, insurance, taxes and duties.
It's the margin than enables a business to build, to acquire land, and infrastructure, 
-not equipment and updated tech because that's amortised against the taxes.
Profit can also be used to put something aside to protect against falls or adversity in the market, and so on.
Better to have that something than rely on insurance - lest the premiums rise beyond affordable.
Without that edge, few businesses would survive long.
Supermarkets, averaging 3% profit, fly perilously close to the wind.
Some other businesses make such outrageous profits that I consider them sharks; I won't go near them.

Charities are the ones that couldn't exist at all unless kind-hearted people cared.

The non-profits are half-way between. Most keep afloat only because they're careful to raise enough to break even.