I'll probably jinx myself, but everything is working now. A few days ago I disabled the Dummy Yahoo account and mail kept pushing, but when I deleted the Dummy account, mail stopped pushing so I added the Dummy account back in and disabled it. This made no sense, but I left it as is as it was working.
After upgrading to iOS 6.1, push was still working so I took the chance and deleted the Dummy account off my iPhone. Push kept working, so my guess is that rebooting to install the update "fixed" my main account.
On a side note, Yahoo informed me that push was broken for many people in December (back when I first had the problem), but this was fixed a couple of weeks ago.