> to clarify, letting the employees of a public corporation decide the direction/priorities would be like if the employees of a non-profit had a union which could force the non-profit to change from planting trees to something unrelated like washing dogs.

Who's forcing anyone? If the company doesn't want to play ball they are free to hire replacements when the workers strike. If the workers don't care enough to risk their jobs over it then their bluff has been called.

If the company does play ball and the shareholders don't like it they are free to elect a new board or sell their shares.