>>The difficulty of objective fact-checking doesn't mean we shouldn't do it. To use that logic, we should also not have judges or a legal system.
The limitations of objective fact-checking means we need to understand where we can/should apply it, and its efficacy are limited.
Judges or a legal system are a good example. These are also truth finding institutions. They are very expensive institutions, and produce a very limited stream of narrow truths. That's because truth finding is, as you say, difficult.
Law is an extremely limited system. We (in theory) apply it as little as possible, when other solutions have failed. The system is set up to verify or falsify claims as narrowly as possible to find this solution. Law has specifically defined truth categories like "reasonable doubt," and other rules about how to lean in cases of ambiguity.
The evidence for "person X killed person Y" may be sufficient to acquit person Z of a murder, but not to convict person X.
Fact checking can be used as a tool that newspaper editors use to maintain quality. It doesn't do the same thing in a different context. Fact checking at a newspaper, in wikipedia or for social media is a totally different thing with different limitations and applicability.
> Let's say you're reporting on Obama's birth certificate, and you write 20 pages of a deep dive into where he was born and all the evidence for it. You are complete and thorough, and you end up concluding that he couldn't have been born in Kenya.
Works perfectly fine for personal fact checking. You can do it like this. If it's being done in court, an editors's office, a wikipedia discussion page or FB's version of mechanical turk for moderators.... you will need a different method.