A related point - The Guardian makes much of the fact that they are funded by a trust fund, arguing that it makes them "independent". I doubt they would make the same argument about other companies or individuals sustained by old wealth.
Obviously that doesn't remove all biases, it doesn't remove all the ways people in the non-profit might still be motivated by agendas either related to wanting to be paid more by the non-profit or to make more money in their other business interests, and it of course doesn't mean they've done the impossible and become a perfect, infallible news organisation.
But it's still a hell of an improvement over the way nearly all other major newspapers are owned.
the answer to this question is well-explained in the post you replied to