With the admissions focus on a holistic process, SAT is but one piece of the jigsaw puzzle. How big a piece of the puzzle it is, will vary depending on the type of university.
For the IVY league and very selective universities, being in the top 10% of your academic class and having the ability to pay full tuition are most important and are critically used to separate one pile of applicants from the other, which however does not mean that a good SAT score is not necessary. The fact is that almost all applicants to these top colleges/universities will have an excellent SAT score, so it cannot be the crucial admissions factor and yes, statistics will tell you that students who are in the top 10% of their class will almost always score well on the SAT barring a few exceptions. Research has shown that students, who do very well on the SAT, do very well in college and top universities are aware of that. They are also aware that good SAT scores of the incoming class translate to a good ranking. So, while getting a strong SAT score is critical it will not by itself get you into these selective universities that will require you to have a special hook, a special talent. The exceptions to the above rule would be the legacy students or underrepresented minorities or the full payers, who might not/need not be in the top SAT percentile range. Also, these universities will expect the applicant to write the SAT subject tests, considered to be a critical indicator of academic success in college.
For the big public universities that have huge applications and very huge slots to fill, the SAT score becomes more relevant and is used as a fast cut of the applicant pool as long as there aren’t any telling shortcomings in other aspects of the application. In other words, SAT scores are used as cut offs to admit students.
And then, there are some universities that in any case are SAT optional, although a student who has given the SAT and has a good score will certainly be at an advantage even at such universities.
Beware of the published SAT ranges by colleges. Being in their published 25th to 75th percentile range and that too in the middle of it, is by no means a guarantee of being admitted. Remember that the competitive students in this middle 50% are closer to the 75th percentile score and thus the aim should be to be way above the mean to get a meaningful shot at admission. The lower scores, below the 25th percentile score, are mostly of the special groups mentioned above.
All in all, SAT scores do play an important role in admissions, just that the degree of importance varies in each case.