Now I'm UTTERLY confused.
I decided to try again on another site I was planning on taking offline and I ran into the same problem. Then I disabled ALL of the plugins and added them back in one by one testing the form each time with incremental subject lines so I was certain I was getting each message.
I reenabled every single plugin I had previously had active and the contact form continued to send to me.
So I went back to the other site and started to try the same thing. Didn't work. Then I decided maybe there was some sort of typo in the email address I wasn't seeing so I changed what email address it went to. It went fine. Switched back to the correct email address and sent. Nothing. Checked the spam folder. Nothing. Checked the trash. Nothing.
a) I have no idea why deactivating the plugins on one site and then reactivating everything one by one seemed to work.
b) I still don't know why, even with nothing else turned on on another site I can't get it to send to the correct email address. Other things are coming in fine to that email account so it's not a problem with that part either.