I've talked to some adcoms and they say it might a bit if they have enough internal data from students matriculating into their school from X,Y,Z undergrad. But usually not a ton.
This isn't a perfect analogy but I've always thought of it as joining a travel sports team for a few years in hopes of making it into the majors. Let's say you have team A (super competitive, nationally ranked, etc.) and team B (local team, solid, but not well known). Just because you join team B does not mean you won't kill it and make it to the majors. Just because you make it onto team A doesn't guarantee anything either. Also the major league teams may or may not even know the difference between team A and team B. However... spending years in team A, surrounded by people who started out at "the top" could force you to perform at a higher level/not be complacent. For example, I go to a pretty intense undergrad and I really think that prepared me better for the MCAT than the ridiculously expensive review course. Anyhoo, just my 0.02