I see the largest internet companies, including Facebook, Twitter, and also Google, fight spam and other harmful content only to the degree absolutely necessary to stay somewhat usable. Which makes sense because it's costly and does not generate profit.
I would expect a non-profit, however, to focus much more on fighting harmful content because it centers around the user experience, hence quality of the content.
I don't see a guarantee this works in practice, but the respective incentives seem clear.
When a for profit entity is more successful at fighting abuse, their users are happier and they sell more ads and so can devote more resources to fighting spam. When a nonprofit successfully fights spam, they don't get more resources, and the spammers upgrade their toolboxes, because they do have a profit incentive.
If you create a search engine where users can report spam and get some form of karma for valid reports that is shown in their social network, then it's quite likely that the users have enough momentum to get ahead of the spam.
There are very few incentives better than profit..This is going to be a US v/s USSR fight during the cold war..