That would require substantial storage, which none of the sticks or boxes have.

AFAICT, an OTA DVR is very much a secondary market; it's not big enough for the big guys to justify building a box with that as its primary purpose. However, it definitely would be a good differentiating point between the available boxes.

But requiring secondary storage is probably a no-go. It's complicated and failure prone.

Which is why the best hope lies in video game consoles, which do include substantial storage. And in fact, the XBox One is gaining DVR-type capabilities in Europe. If it is successful there I hope we will eventually see it in North America too.