I'm thinking that the website should be strongly devoted to neutrality or objectivity, as is Wikipedia.

In Wikipedia part of being objective means to accurately report what's generally known about a topic and not engaging in original research. I think you will have a hard time to judge fallacies without people engaging in something like original research. Fox news has it's own brand of fair & balanced with isn't exactly the same thing that most people think of when they hear the phrase.

What kind of objectivity do you want for your website?

The status quo bias of one person is another person's Chesterton's fence.