I was following all this and all of a sudden realized I had a DirecTV AM21 (<$100) that I forgot about. It does all this...if you have DirecTV. We used it a lot a year or so ago but haven't recently Only time we use it is we can't see the bird because of trees etc so it works perfect...for us. Your results may vary.
On Edit: Nevermind. You gotta see the bird to download the OTA channels so this only works if you're in one place. Maybe if you knew where you're going and downloaded while you can see the bird it would work. Back to Walmart for the $50 one.