I agree with you that the current system is counter productive and I would even say that it is worse that the Hattricks system. If we look at the Swedish league their are at least three teams left that are left in the last 16 based on (almost) pure luck. I actually had a pretty descent draw my self, not particularly lucky nor unlucky, but still the complete randomness of the drawing makes me feel that I actually have not been accopmplished anything by reaching this stage: Advacing in the cup should feel as an achievement and not as luck!
The problem with the Hattrick system is that you are not really rewarded if you accomplish something big (i.e. if you beat a higher ranked team), as you will end up meeting yet another top rank team regardless of your achievement. Therefore the system you propose is far better than both Hattricks and BBs current system.
With that said I actually prefer the classical seeded drawing that is used in most sports today. I.e. you seed the teams and then make a drawing with a complete playoff three based on this. Then you will get rewarded for accomplishments, while "pure luck" will be minimized.
As an example if someone upsets Roger Federrer in the first round of Wimbledon he will (in some sense) benefit from this as his next oponent will most likely not be someone like Nadal, in fact Nadal will most probably await in the final if that "someone" manages to advance there too.