Yes, I think the point was already made up-thread that it’s best to start out in engineering if the desire is to keep that door open, but it’s good to reiterate that point and elaborate on the multiple reasons why that’s the case.
For a student who might pivot to business, Industrial/Systems Engineering could potentially be a good place to start. That would keep the student on track in terms of core engineering requirements, so switching into MechE wouldn’t be as difficult as from a non-engineering major (though as you said, how impacted different majors are may vary depending on the school), but IE is also more business/econ adjacent, so it can offer a good middle ground to be able to pivot either way. Depends on the program, though; it’s good to look at 4-year plans for all majors of interest, to determine the extent of overlap and where/when they diverge.