Applying those standards would rule out most historical evidence for everything that happened in the pre-modern world, and many things we know about the modern world too.
If we set ourselves the standards of rejecting everything but facts as evidence, we'd be much worse off than using a standard of the careful and critical evaluation of evidence.
Sometimes, it's just not possible to provide empirical evidence for claims.
But when that is the case, we don't just simply believe whatever the "testimony" says.
Instead, we give it a degree of reasonableness and believeability.
We try and correlate independent, preferably contemporary, "testimonies" with eachother and cross reference.
For example, if two opposing factions of a battle have independent records of said battle, and they both agree with eachother... then it's quite reasonable to conclude that a battle took place and that that's likely how the battle happened. If they both have totally different accounts of said battle and there's no empirical evidence of it and say even the timeframes don't match, then how are you going to decide which one is right? Was there even a battle to begin with?
And if you then have even only a as-good-as a single source mentioning it, what then?
Then you look at the kind of claims it makes. If the claims are compatible with what we already know, then that is "advantage believability". If not, then the claim is rejected (and how dismissive we are is usually in proportion to the extra-ordinarity of the claims).
For example, that's how we have less problems believing Socrates was real, as opposed to Atlantis even though we (basically) only have Plato talking about either.