In some ways, I think this question acts as a good determiner for whether you treat the pasttime more as a cooperative story or as a game. Game folks, I think, prefer the charisma-only intimidate; folks that approach it as telling a story together prefer the flexible intimidate.
That said, I'd only allow it in certain, logical situations. Whereas the charisma-based intimidator might be able to scare a prison guard into letting her go ("You can let me out now--or I can escape sometime during the night, slit your throat, and deliver your severed head to your loving wife, and can't you just imagine her screams?") the strength-based intimidator isn't going to be able to do much ("Uurgh! Og smash!" doesn't impress a gaoler very much).
But in a barfight, things might be different. The 22 str barbarian wants the fighting to stop, so he picks up a table and snaps it in half. People might not choose his side of the argument, but they sure start paying attention, and the ones that judge him more powerful than themselves are likely as not to stop fighting. They may run away, they may start trying to pacify him, they might shout for the guards--but they'll stop fighting.
I'd allow it because it's fun. Heck, I'm the charisma-monkey in our group, with a feat and a racial bonus on my charisma checks, and I still love watching the party tank intimidate people. It leads to a better story for us, leads to a better time for us.
Does it bend the rules a bit? Sure. But the rules are there to serve the stroy, as far as I'm concerned, and when a rule gets in the way, it gets bent.
Daniel