Back when Java didn't have lambdas, one of the more advanced lambda proposals (http://www.javac.info/closures-v06a.html) had this exact thing for this exact reason.
Unfortunately, this take on lambdas was deemed too complicated, and so we got the present system which doesn't really try to deal with this use case at all.
Well, scala has union types, but it doesn't do checked exceptions per say (but it does have a very advanced type system so similar structure can be easily encoded). I think checked exceptions is pretty rare, so I don't know.. probably some research language (but they often go the extra mile towards effect types)
Of course, if you had proper sum types, that situation wouldn’t be a problem.