Note that a for-profit owned by a non-for-profit is really a not-for-profit. Because the only things that the for-profit company can do with it's profit are reinvest it, or give it to the non-for-profit.

>Because the only things that the for-profit company can do with it's profit are reinvest it, or give it to the non-for-profit.

It can also pay it's Cxx's nice salaries and hefty golden parachutes, which makes it a for profit for them...

Then what is the benefit of this arrangement?

That is, based on what you said, it sounds like there is no meaningful difference between having a non-profit do all the work, and a for-profit doing work on behalf of the non-profit that owns it. But if that was the case, why would anyone bother with the extra layers? I'm inclined to assume there is some reason for the extra layers, and I have difficulty believing those differences are in society's best interest.

I think the difference is mainly in restrictions on what kind of activities a non-profit can participate in. I'm not sure on the details.

Taxes are a big reason. In many countries you can save quite a lot on revenue tax this way.