Is a broadening assignment absolutely necessary in order for an E6 to make E7?

Technically, no. Statistically, yes.

Yes. I think its something the look at. You need to make yourself stand out and just being in a company doing your normal job won’t do that. Drill Sergeant, Instructor, a degree, airborne air assault are things you will need to stand out among your peers in your MOS. I am in a really small MOS and we don’t have many opportunities to make 7. Most of the ones that have have done something like drill sergeant or recruiter.

I once feared the drill/instructor/recruiting assignment. And when I was in a competitive MOS, it was hard to separate yourself (insane OPTEMPO in CA) and then I got put on Recruiting, and like every other 38B who ends up Recruiting, got picked up for 7.
Not only is it an assignment to truly separate yourself, but it's the chance to give back to the Army. You as the DS get to shape the new Soldiers.
Broadening is good, very good. But there is more than just DS out there as well....MUCH more that branch managers and retention don't know or won't talk about.