Yes, I’ve got a similar issue, although the emails do appear in my inbox.
However, I get no notifications of new email (e.g. on my Android phone), so I have to explicitly check regularly to see if I’ve received any messages via my website.
Other than that, all works really well, but would love to find a solution to this problem.
I suspect it’s to do with aliases, etc.
I wonder if it’s worth setting up a separate email address @mydomain.com purely for acting as the From address for messages from website.