in the past on several occasions ive seen hiring managers open to interviewing candidates with no relevant experience but simply because of the brand name. the brand name elicits a signal of quality that is hard to ignore (consciously or subconsciously) - that may change in the future but to this point my personal opinion is going to harvard, mit, yale, princeton, columbia, etc. is absolutely an advantage in getting "noticed".
imo paying out of pocket for an MBA from a mid tier or lower school is increasingly less useful, bordering on being completely neutral in terms of benefit while having substantial opportunity cost. certain scenarios (employer paid, necessary to break through glass ceiling, etc) can certainly change that calculus.
my personal experience (undergrad from a mid tier expensive private university in a large city, MBA from a state school) is that almost no one has cared about me having an MBA to this point, and the brand names of either school are nowhere near strong enough to get noticed. my MBA was a very unique program and I feel was very enriching but unfortunately others can't see that in a quick glance.
if I had to do it over again i would attempt to get a stronger brand name somewhere in my resume - whether undergraduate or graduate. in fact im considering further graduate study for that exact reason.