I think you need to look at this from a very analytical perspective regardless of who is paying. If you had $260K, would you spend it all on Amherst or go to UMich and have $165K left? Wouldn’t that money be better spent on graduate school? That will buy you an MBA, a law degree or 3/4 of an MD. Would you take on $165K of debt for a Liberal arts degree? I wouldn’t. Could you negotiate with your parents to set aside the savings for graduate school?
I also think you are confusing selectivity with quality. The relationship is not direct. UMich CoE is one of the 10 top engineering schools in the world. Ross is in the top 5 for undergrad business. Either of those degrees are marketable and worth real money. Both are much harder to get into then normal UMich. You also have the good fortune to be in-state and pay incredibly cheap tuition for the quality of education you will get. Median ACT for UMich CoE is 33. For Amherst it is 32.
If you were purely interested in Liberal Arts, compare the UM Residential College with Amherst. The RC gives you the best of both worlds. You could also do one or more - LA/Bus Eng/Bus etc which would likely take an additional year. Both CoE and Ross are small enough that you won’t get lost. Also look at the honors college for LSA and engineering. Did you apply to be admitted as a Freshman into Ross?