undiscriminating skepticism

I think that's a bad description. The kind of people on RationalWiki are very discriminating. When something is said by an Authority they trust they aren't skeptic and when something is said by someone they don't trust they are very "skeptic".

Maybe we could have multiple anti-bullshit websites

I don't think that framing yourself as "anti-bullshit" is helpful. It's makes more sense to frame yourself as being pro-evidence. We already do have multiple websites that do explain issues.

I personally like Skeptics Stackexchange. If I come about a new claim I often simply go and open a question over there.

When it comes to an issue such as vaccination I think Vox has a decent primer: http://www.vox.com/cards/vaccines/what-is-vaccine