Mythos, from Flagship Studios underwent a Beta process much like the betas described by Stardock. The game in its initial form was very primitive and did not have a lot of features. The entire point of Mythos at that time was to test the netcode (and to a lesser extent, the Engine) for Hellgate: London. As it grew, more features and gameplay were added for the players, but the process was very much about finding bugs first and worrying about feature addition and balance later, as the game progressed, though major funbreaking imbalances were fixed earlier so the beta testers would at least enjoy the product they were testing.
I think Stardock's Beta process is a fine method, but I don't think the goal of the primitive forms should be to be "unfun". If you're intentionally stripping the potential for enjoyment out of your game for the sake of calling it a beta, then you're intentionally making the process unnecessary tedium for your testers.
Yes, there should be a bit of tedium in the hunt for bugs, but it shouldn't be artificially forced, nor should it be overlooked when players are expressing a point that makes the game unfun for them in the beta process on the grounds that "it's just a beta."
Regardless of the status of the game, a feature being unfun is still unfun. Having people search for bugs in a barren form of your game is fine, and I understand that it helps you find the more important bugs first (since there's less fluff to sort through to find engine bugs)...but if your gameplay at that point has problems that discourage people from trying it, it's a problem. Sure, the game may not be for that person. But on the other hand, by devaluing or even ignoring that tester's opinion on a gameplay-related feature, you may be losing one of your early testers, and the process is bound to repeat itself.
Betas don't have to be fun, but the game being tested does. And if it's not fun in even a barebones form, something's missing from the game.