In most places, the scale of the problem results in this being a largely clerical problem. Most colleges and universities have an office responsible for advertising programs, processing, and accepting or rejecting undergraduates.
The "admissions office" likely hires counselors, and sometimes students, to interact with secondary school academic counselors and with individual students.
One reason for this is that it is seldom necessary to declare a "major subject" during application in the US. The education itself is quite generalized with everyone studying broadly. Thus, a math department, say, doesn't really get involved, though it might be occasionally asked concerning some specific student.
As a postdoc you would be very unlikely to ever be in the loop and certainly not as a job requirement.
Note, however that someone at the university needs to set policy for the "admissions office". Faculty might be involved in that, but it is more likely to be tenured faculty who have some experience with the institution. Not guaranteed, but likely. Good universities also tend to want to "clear the decks" for untenured faculty to give them a chance to learn to excel at the institution.
