I only have knowledge of American universities: I know of junior faculty that serve on graduate admissions committees in the STEM fields -- at top schools such as Harvard. The things that I hear from them are quite interesting, for instance, how strong the recommendation letters are from the undergraduate applicants at top schools - that virtually all letters say that those students are the "best" that those professors have ever had. But why do junior faculty such as "Assistant Professors" or "Titled Instructor" serve on graduate admissions committees? To me, the admissions process for graduate STEM programs are so important that I feel the committee should consist of senior faculty instead.
If junior faculty members aren't allowed to sit on an admissions committee, then by the time they become senior faculty they won't have had any experience of sitting on admissions committees. The junior members need to learn how to become senior members and this is just one part of it!
I also don't think that graduate admissions decisions are quite so monumentally important (or tricky) as you imply; for top institutions such as Harvard you will already have a self-selected pool of very good appplicants to choose from, and I imagine it's difficult to make a bad pick.
As an applicant the decisions of the admissions committee are critical, as a faculty member, they don't really matter as long as you get it close. In the worst case, a school passes up a future Nobel Prize winner for someone who fails their quals and life of the department goes on. In contrast, Assistant Professors also sit on faculty hiring committees where a mistake means another faculty search and possibly the loss of a tenure line.
As for why they sit on the committees, there is a ton of admin work in a department, and some of it truly needs to be done by Full Professors (e.g., tenure and promotions) and you need to use people where they can contribute.
In addition to other excellent answers, I would add a pragmatic (and even cynical) viewpoint: serving on admissions panels includes a lot of boring paperwork which is not visible or recognised outside the university (e.g. does not lead to peer reviewed publications). Since the currency of academia is fame and prestige, senior members don't want to spend their time on admissions panels (and they have a lot of other administration duties to take care of). Since they usually have enough gravitas to say no, the unprestigeous task inevitably falls on younger colleagues.
As other people have said, tenure-track assistant professors have to do a certain amount of departmental service to get tenure, and being on the graduate admissions committee is a good way to do this. Lots of committees require work thoughout the year, which ends up taking a lot of time. Grad admissions basically means that you have to spend a day or two reading files and then attend a couple of meetings to hash out who gets admitted. High impact, low commitment.
One thing from your question that hasn't been addressed is the fact that at some departments grad admissions is handled by "Titled Instructors", who are not on the tenure-track. At least in mathematics, this does happen at a couple of places like Princeton (where the senior faculty are particularly effective at avoiding departmental service and which doesn't really have a tenure track), but it is very far from the norm. In the vast majority of departments (including fancy ones like MIT and Chicago), non-tenure-track faculty are not expected to do any service at all.
TLDR: A combination of self-interest and the pigeonhole principle.
As a junior faculty member in the sciences, I've been advised by a few mentors to be on the grad admissions committee. Junior faculty need to build up their research groups, and the graduate courses they teach will be populated by the incoming students. Choosing students who could fit in those two roles is highly valuable to a new faculty member - but less critical to well-established groups!
In addition, it is to some degree unavoidable. Everyone does some service work in a department - this can be acting as chair, serving on university-wide committees, reviewing tenure cases, organizing seminars, redesigning curriculum, hiring faculty, dealing with admissions, etc. Some of these are restricted from junior faculty (e.g. no one without tenure reviews tenure cases or chairs the department, I think), and others are merely inadvisable because they require huge time commitments and better knowledge of the department (curriculum redesign). This means things like hiring committees and admissions committees are often mixed junior/senior groups - there are not enough senior people to do all of this work alone!
In addition to the other excellent answers given, there's also a number of other considerations:
- In programs with funding coming primarily from grants, programs with lab rotations, etc. these junior faculty will be expected to host and mentor these students as well as potentially serve as their advisors. How receptive do you think they're going to be to "Hey, you had absolutely no input in this decision, but this student is going to need to hang out in your lab for the next six months"?
- Junior faculty can represent new directions for the department. For example, my department is currently hiring a number of more theoretical/computational faculty. They will all be "junior faculty" - and yet the department is hoping to recruit students who will work with them. Would you want to be evaluated by an admissions committee whose entirely unfamiliar with your field, qualifications, and what would be expected of you? Ones with entirely different expectations of what should be "required"?