I have heard that authorship of the book of Ephesians is contested, and so might not actually be Paul. The point is that if it is not Paul (and in some lines of thought, even if it is Paul), it is not divinely inspired and so does not represent the Word of (the biblical) G*d.
I think it is more relevant to consider what Jesus was recorded to have said about the alleged directive for women to be submissive. What did Jesus say about the need for women to be submissive? And if Jesus said nothing, why did G*d rely on surrogates to convey such important information which would then if true govern such a large proportion of followers of Jesus?
None of the gospels are attributed to Paul, and none of the books of the Old Testament were written by Paul, so why would Pauline authorship be crucial for Ephesians to be inspired?