I don't understand there to be anything deeper about it, though happy to be corrected. In a consistent system, there are three kinds of statements: the provable ones, the disprovable ones, and the independent ones. This is straight from Godel Incompleteness I. The sin of LEM is exactly that it overlooks the reality of incompleteness. But, if course, you also can't fix it with an axiom encoding the trichotomy I just stated, because the independence of a statement is itself a statement that lives in a higher ambient system.
LEM doesn't overlook completeness within the context of an axiom system. Independent results remain, by definition, without correspondence to any proof.
Believe it or not, constructive math actually does have a LEM, but a derived one. Any computable value can be shown to either equal or not equal another. Only for uncomputable predicates does the absence of LEM result in our inability to conclude x from not not x.
