Danny, Danny...I know what the rules say. And, when the rules don't match the thing they are trying to model, you have a disconnect. Thus, this thread to talk about it.
There is no problem with the rule. It's your grasping the game's level of abstraction.
You keep talking about a low Cha but intimidating character. Someone who is more of a follower and has low force of personality, but is nonetheless scary. RAW, the only way that happens is if he takes ranks in Intimidate. There simply isn't a low Cha & no-Ranks Intimidating person in the game because the rules are very clear about what "Intimidating" means => "reasonably assured of making a successful Intimidate roll for his level."
In D&D, your Cha score irreduceably combines a wide array of mixes of attractiveness and personality, and you can't simply pull out one element and go from there.
But, logic tells you that the Barbarians should have an intimidation factor that is exactly the same,
No, the clothes don't make the man. Nor does the class. The man is the sum of his parts.
Like I said before, you dress me up like that picture, and I guarantee you I would not be intimidating. Laughter would be more likely. My stick-figure buddy would also not be intimidating dressed in that fashion.
That has no bearing on whether or not either of us could kick someone's ass. We're simply not much to look at, in terms of fearsomeness.
To illustrate further: there was a heavy metal bar I used to go to regularly that had 2 bouncers. They dressed identically. One was 6'3", 330lbs and had a Mohawk, and a booming voice. Very intimidating. The other was 5'5" and about 145lbs. Not much to look at, fairly soft-spoken. In D&D terms, they had the same "class", and they wore the same clothing. They were not equally intimidating.
The big guy's visual impressiveness kept bad behavior to a minimum, but when things got tough, they got Mr. Lee. It was
his job to clear the bar of idiots, which he did by kicking people in the head and letting the big guy drag them out...