Those are grounds for a skeptic to reject such claims. That is not the same as calling them wrong. It's refusing to call them right and therefore remaining agnostic on the matter, or ignostic in the case of the supernatural, since nobody seems to be able to say just what the supernatural is, just what it isn't, which is presently indistinguishable from the nonexistent..
Then you should be skeptical when I make a claim about what that was.
That's not what archeology does. Not much of my day or yours from a year ago can be dug from the ground.
Skeptic is based on the assumption that the claim doesn't affect your life. The the claim that "there's a bomb" will leave you with no middle ground to be neutral to stand skeptical. You either believe the claim to run or to reject it by believing that it's a hoax. Christianity claim is such a claim. We believe because eye-witnesses sacrificed their own lives to stand witnesses for Jesus Christ.
Archeology is the study of scarcely preserved location which may leave you with some clues of what could possibly happened. It usually requires,
1) a claim from a human writing, so that we us archeology to try to verify the claim
2) there's a large scale human activity happened and recorded down. Usually, it is a city ever lived by mass of humans. Or a war event with mass of humans involved. It is because only so a trail could possibly leave behind. It never applies to the deed or speech of an individual figure. You can't use archeology to prove what George Washington said. There's no trail left for individual deeds and speeches.
3) the location for some reason is reserved. Say, an earthquake buried a city which is discovered today.
The rest is up to the archeologist to interpret what the contents preserved are.
You can randomly fetch any history book written more 1000 years ago (in order to be demonstrative), then to go through section by section with the question, "how this section supported by archeology"? Then come back to tell us how many sections out of the total sections of such a history book is supported by archeology. What archeology can prove is minimal.