Well, let's take on our "verisimilitude" hat on. Should there really be a way to play a hulking creature wielding a two-handed axe larger than yourself that wrestles polar bears and dragons to death and can probably rip out your arms out if he wanted that is not intimidating? I mean, he got the muscles, he got the rage, he got the kills, he good the blood stains and scars.
Not really. Picture him played by Barney Fife, but with muscles. Do I have the ability to incinerate him with a fireball? If so, why would I find him intimidating? I would consider him dangerous. Dangerous is not the same as intimidating. Guns are scary. If I hand a gun to my four year old son, he becomes very scary and I become scared, but that doesn't make my son intimidating.
If a character is simply scary and that causes people to run away or try to kill them, that's not intimidation. That's people making the expected, rational choices. Intimidation is used to convince people to do something other than what their natural reaction would be. In
Willow, Willow tries to intimidate someone with a magic acorn. It doesn't work, even though we later find out that the acorn is terrifying powerful. People just don't believe him. Intimidation is John Constantine staring down gods and demons, with little more than hunch and grim expression, and making those eldritch things hesitate. Intimidation is "The Gambler" holding a guy at bay with a gun that may or many not be loaded, or the Dread Pirate Roberts starting down Humperdinck while all but paralyzed.
For a hulking barbarian to be intimidating, he would have to be able to cow a city council into meeting his demands (rather than scaring them into hiring a bounty hunter), cause wizards to quake in their boots (rather than reaching for the sulphur and bat dung), and sit down at a game of chess and make the other person briefly terrified that the barbarian is actually a chess-master underneath all that polar bear fur and scars.
Intimidation is a short, unassuming guy starting down that barbarian without even blinking, and the barbarian halts because, if that guy's not scared, why not? And it certainly doesn't depend on Strength. An orc is generally not at all intimidating, because I would never be tempted for a moment to think I can appease the orc. My impression of the orc is probably that he is stupid, dangerous, and unsubtle. Therefore, while I might be afraid of an orc, if I were about to have to fight him, I would not be intimidated, because my first reaction would not be to be cowed, but to figure out some way to kill it or escape. If I meet an orc in the marketplace, I might be somewhat intimidated if we get into an argument over the last pie, but the orc's Strength is probably less of an issue than my belief about whether the town watch is nearby. A soft-spoken halfling with a razor could be as intimidating, or more intimidating, in such a situation.
Are cows intimidating? Dump trucks?
The disconnect comes when someone expects their hulking axe dude to not merely be scary, but intimidating, based on their image of the character. They want an ability they actually have to budget for. Once your Intimidate is maxed out, it hardly matters what your Cha is, and if you want to be truly terrifying, take Daunting Presence of Skill Focus (Intimidate) if you are playing 3e. The reason this is so is because it is very easy to imagine a character who is strong, deadly, and wielding an axe who is basically not intimidating at all. He seems like an oaf... until we have evidence of his terrifying prowess, he doesn't seem like a threat, he seems like a yokel with an axe.
A silence spell isn't intimidating, but it could be terrifying for a wizard. That's why singling out Strength is just poor game design. It is only one of several possible factors that might apply. However, Charisma always applies. If you don't convey at least some sense of confidence, you will never be as intimidating as someone who oozes self-aware menace.