Sure. Although I'd say that if you want to study history that's _all_ you can do - use different sources, corroborate, cross-check, link and, generally, try to make the different events and interpretation "fit" together. If you have no documentation for it or supporting evidence then you've got nothing to work with.
Otherwise one could just use a semi-apologetic argument: the Exodus story DID happen as outlined in OT but God hid all signs of it so it couldn't be confirmed.
Sure, but I think 1) a lot of objections in this thread come because people seem to conflate "nothing to work with" and "so obviously it didn't happen" and 2) there's not no documentation for anything people have argued about in here.
A big disagreement that's probably been unsaid in this thread has more to do what counts as corroboration. Speaking abstractly, I think that if a group of people from 500 years ago strongly attest to something that happened 1000 years ago, that is not definitive proof in and of itself, but it is absolutely a form of supporting evidence.