Do you need to have business-stuff on your resume?

I apologize if this question is sleep-deprived and stupidly worded.

Let’s suppose, hypothetically, that as a mechanical engineering major, I earned a 3.8+ GPA, a ridiculous score on the GMAT, and have ample work experience/personal statements/etc…but it’s pretty clear that they all relate to engineering and not to business or management (beyond perhaps incidentally receiving leadership positions), and it’s almost as if I applied to HBS just as a crapshoot “if I get in I might consider it” plan.

I’m guessing Harvard would throw my application out. Which business schools, if any, wouldn’t?