I think it's more useful to have some kind of system where we can make more reliable determinations. They won't always be perfect, but there should be ways that we can separate claims. Going by what we want to believe would leave us with an endless list of claims for almost every possible event from the past. And it seems to me that historians are not that scattered in terms of what is considered reliable or not.
I think our personal experience does help, if we take into account how we deal with such claims in our day-to-day lives and in other facets of our lives. How willing are we, normally, to lend credence to claims that have little to no corroboration? Do we approach similar claims differently based only on which ones we favor? We should be able to establish a pretty comprehensive sets of norms that most people can easily agree on.