If TIVO ROAMIOS can have 6 built in tuners, and Directv Genie can have 5, I don't see the complications. I think Dish keeps its equipment limited to force customers into adding more receivers, which adds more fees.
It is a physical limitation on the way DISH stacks satellite polarities on the feed to receivers.
On a single polarity feed one can tune any transponder on that polarity.
On a stacked feed one can tune two transponders (theoretically one could tune two or more transponders from the same polarity and additional transponders from the other polarity, but in practice DISH uses the stacked polarity for the second tuner).
The feed to the hopper is triple stacked ... three 500MHz bands with DISH tuning one transponder from each band. That limits DISH to three tuners per cable. With PTAT activated DISH uses one tuner for four feeds, which gives them the ability to record six channels at the same time.
DirecTV's single wire multiplex stacks in a different way ... instead of stacking the entire 500 mhz polarity they stack transponders and assign each to a receiver on that cable. A five tuner receiver uses five slots. The Tivo you mention is for cable where they don't have satellite polarities to stack.
It is easy to go for the conspiracy theory attack and say it is for more fees ... but it is just a limitation of their technology. Perhaps they should have gone for something more like SWiM (if the patent allowed) but they went with their own stacking scheme.