I don’t think there is any reason to believe an application coming from Bing, Geneseo, CUNY Queens would be regarded differently from an application from Vassar or Haverford.
However, the environment and the opportunities at each will be quite different.
For people who care about SUNYs, Bing, Geneseo, Albany, and SB are the 4 “academic” ones. In keeping with the SUNYs principle that each college has an area of unique strength (while being good elsewhere), Albany is known for Business and social science, SB for STEM, Geneseo and Bing for traditional liberal arts in 2 different environments (LAC and large university), and Bing additionally for its very selective School of Management. Buffalo has pulled up in addition to these 4 by focusing on Engineering and doing an excellent job there. The CUNY system is respected, too, but it will not be the residential experience you’re looking for and you can choose between two stronger options.
Geneseo will have more personalized pre-med advising (probably no pre-med advising available at Bing till 2nd semester sophomore year: advising time is tight, no sense in wasting it on students who may end up with a C+ or B- in General Chem. May seem harsh but tuition is kept low by rationalizing everything.) Geneseo professors will also be more likely to involve you in research, since you wont be competing with grad students for access. On the other hand, Bing will have more research opportunities overall, especially in terms of large or nationally-funded projects + especially if you’re in Honors and get to know your professors better through smaller classes. There’ll be more clubs and groups, more sections of the same class.
Now, if you get into Vassar or Haverford, it’s a whole other kettle of fish. First, because these are literally life-changing, unless you’re already upper middle class. (The term “life-changing” used in a study that indicates this type of colleges is specifically for working class or lower income kids, children of immigrants, first gen college students, and URMs). It’s not about whether attending Vassar will make it easier to get into med school but for everything else, that is so intangible it’s taken for granted by kids from upper middle and upper class backgrounds (also, note that between Vassar full pay and Bing at 1/4th of the price, they still choose Vassar overwhelmingly so there’s got to be a reason, even if for them it’s not, in fact, “life changing”.) Your mother is offering to pay because she lived it, personally, and knows from experience what I’m talking about.
Second, it’s quite possible your FA package will be better from one of these colleges, especially since SUNYs package loans, even with Excelsior.
Third, because there’s grade inflation at some private colleges. And no, the algorithm that makes the first cut doesn’t adapt to the college. A 3.65 from Brown (barely average) is treated the same as a 3.65 from a college where the average GPA is 3.1. (The reasoning is that the students are probably more accomplished and competitive at the famous private college where the average is high. Or simply the first cut algorithm is not meant to do more than eliminate the clearly unqualified, which 3.65 wouldn’t be anywhere regardless of average GPA at the college.)
Finally, most students who intend to attend med school change their minds along the way or don’t make it into med school, so your choice has to balance factors in case you do apply and in case you don’t.