If what Senzaemon said was "Please stop Azami from destroying Tootsuki" sure, I'd agree with what you wrote.
But what Senzaemon asked Soma was "Please save my granddaughter".
Soma might be the most natural born, happy go lucky leader type that can rally everyone on in the world, and it would still not make sense to ask him to help Erina rather than asking Alice, the one who regards Erina like a sister, to help her. It doesn't matter if Alice is an air head, when a family member she loves dearly is in trouble, she would jump in and help. Like she did.
Sure, we could say that the resistance started thanks to Soma, and that in turn allowed the others to step up and do their duty, but it doesn't help with Erina's spefic problem with Azami. You point out that Soma if friendly with almost all the first years. That's true, with the exception of Erina who up until Soma got 2nd at the end of the tournament couldn't stand him at all. So why go and ask the one guy his granddaughter hates to help her?
Senzaemon asking Soma still makes no sense. But thankfully he refused and we got Alice doing what she can, so things worked out.