FWIW, I don’t think there are simple “Cut and Dry” answers to your questions. If you’re smart and work hard you’re going to get where you want to go regardless of whether your undergraduate education is perfectly “lined up”.
Remember, too, that you think you know what you want to do now. That can, and likely will, change. That’s where I hedge my bets.
FWIW, I seem to recall that in the field of accounting, UIUC is consistently ranked at or near the top. But, then again, it doesn’t matter because the best place to start off after college in that career path is at one of the Big 4 (5 if you count Moss Adams), and, believe me when I tell you this, it really and truly doesn’t matter where you study accounting. Unlike the law, the accounting world really just doesn’t care … there are kids at Seattle’s E&Y office with accounting degrees from Western Washington University … a lot of them.
But, as I said elsewhere, if you want to be recruited by Goldman, JPM, or one of the consulting firms, then perceptions matter, and prestige and selectivity matter.
If I were in you shoes and thought I wanted to be an iBanker, but also aware of the reality of MiddelburyDad2’s advice, I’d go to the top school among my choices that I could afford, and by “top” I would not focus on this or that department too heavily. I would also not major in business as an undergraduate. I’d major in econ and/or math. That’s the path I’d take based on what is approaching a long career in corp. fin., law, M&A, and private equity.
PS: there are also schools that are great at getting kids placed on Wall Street outside the Ivy League, Stanford or Duke. Lehigh and BC send tons of kids there, and of the LACs, Middlebury, W&L, Davidson and Williams send quite a few also. Of course, Vassar, Bowdoin, Amherst, Wesleyan, Pomona, Claremont, Carlton and others will get you there too, but my very general observation is that those kids tend to want to go on to other things. My litmus is, “where does Goldman interview on campus?” I know that firm very well. They are not an easy group to access. If Goldman interviews on your campus, that will tell you quite a bit, because they don’t get more particular than Goldman Sachs.
One more thing. This comment, “because UT Austin is more known [than Dartmouth” in business," is not true. There are people at Tuck who would take issue with that comment. 
No, not at all the case. Tuck was founded 117 years ago as the first graduate school of management. No, whatever Texas has to offer, it’s not “more known” or better than Dartmouth in business.