There are no other places the reasons for the wage gap could be hiding.

Again, I'm not arguing either of these. I'm not saying it's something else either, I'm saying overall, it doesn't matter, because whichever you choose has no more value.

You are contributing more value in a higher paying job. If teachers were valuable they would be paid more.

This is not true. A teacher is more valuable than a small grocery owner. Without the small business owner, nothing changes. Their customers can go to Wal'Mart. The reason the teacher is paid less is because more people are willing to do it than start a grocery store. Easily replaceable =\= a lack of value.

This is because the compensation you receive is not directly related to how much you're contributing to society.

And even then, your entire sentiment is that it's inherently better to be of higher value to others. If the shoes are being filled, is it better for you to be the one filling them? If you lived in a small village in the 1800's, would it be better to be the blacksmith or not need to be, because there's already a skilled one there?