There are some things non-profits are not allowed to do. But, owning a for-profit isn't one of them, and the for-profit is allowed to do those things. So hence this is a common strategy.
Example: Suppose I bulk buy T-shirts printed with my cool logo for $15 each and I sell them to consumers for $50 each. That's a for-profit activity, if Walmart was allowed to have a "non-profit" arm which did this I'm sure they would, the tax saving would be considerable.
That's decidedly not what "nonprofit" means. A nonprofit can absolutely fundraise like that, and pay the employees above market salaries, or they can use the funds for other things, they just can't return them to shareholders (of which there aren't any), or the board.
Example: Suppose I bulk buy T-shirts printed with my cool logo for $15 each and I sell them to consumers for $50 each. That's a for-profit activity, if Walmart was allowed to have a "non-profit" arm which did this I'm sure they would, the tax saving would be considerable.