You are confusing logic and meta-logic. In a a suitable logic (per Gödel theorem), p or not p is provable. Only the meta-logical statement "every proposition of the logic can be proven or refuted" is false. This doesn't make the principle of bivalence meaningless, but simply illustrates the limits of the axiomatic method. Intuitionist logic doesn't help at all since Gödel theorem apply to it as well.
There might be reasons to be intuitionist but Gödel theorem is not one of them.