LUL- As far as the specifically Ivy League colleges, you can’t really just get in with motivation and good SAT scores. But that’s true no matter what school you come from, though people who went to the best public/prep schools do have a better chance- more AP classes, more clubs, more leadership opportunities, all of which Ivies will look to. But even people with the best records compete to get in, and if you can convince them that you have something about you that stands out, then you have a shot. But like you said, that can apply to anyone from any school.
For grad school (especially law), yeah, you need good scores. I know a guy who just went straight from a very RW yeshiva (with his last secular education having been in eleventh grade) to Harvard Law School with good scores but needing a LOT of work on writing and other foundational stuff. Not that he won’t do great, I’m sure.
Of course, if you’re trying to get into most other colleges (like CUNYs, Rutgers, YU, Touro, etc- probably NYU as well, though it’s a bit more dicey), the educations you’d get from basically everywhere on this list would get you in and you’d do very well.
Okiale- one thing- you say you’re looking for a yeshivish place, and your parents are looking for college prep- I’m sure this is something you’ve thought of, and the two aren’t necessarily mutually exclusive, but have you and your parents given thought to what your trajectory will be after high school? Many yeshivish schools encourage several years of beis midrash learning before college, or doing college part time while simultaneously in beis midrash, or even going to Israel. Where are you on that, and are your parents on the same page?