Hmm. If I were to take a statistical look at it, I'd say the evidence shows that I prefer stories that have no Big Three (or at least very little of them). Here's my list of stories I like that have no Big Three (or little of them)
KOTOR (and comics)
Republic Commando Series
Forced Unleashed II
And the list of Big Three stories I like:
Shadows of Mindor
The Thrawn Trilogy
The Thrawn Duology
Some books of NJO (Can't remember which ones, so I'll leave it vague)
Fate of the Jedi
When you put it like that, then I should like them more without the Big Three, but, in truth, I have no preference as to characters. As Nedara said, a good story is a good story, irrespective of who's in it. Sure, Luke, Han and Leia are better known, and, when I read them, I don't think 'oh, that's clearly a Han Solo rip off'. I have absolutely no problem reading a story with non-Big Three characters.
As to the Big Six: If it was to be expanded into such a thing, should it not be more prudent to make it the Big Seven? With Ahsoka?
I am a Star Wars fan. That doesn't mean that I hate or love Jar Jar. That doesn't mean I hate or love Lucas, or agree or disagree 100% with him. That doesn't mean I prefer the PT over the OT, or vice versa. That doesn't mean I hate the EU, or even love all of it (or even read all of it). These are not prerequisites. Being a man is not a prerequisite. Being a geek is not a prerequisite. The only prerequisite is that I love something about Star Wars. I am a Star Wars fan.