The application committee is a group of about 10 students and 2-4 staff and faculty. By the time of the interview, everyone will have read all parts of the applicants materials. Close reading and critique still characterizes much of the interview. But in my experience, someone can have major mistakes in the essays/tests/transcript and be admitted. Sometimes the application is deeply rooted in a specific intellectual tradition; sometimes it may ask questions like "which would win in a fight, a bear or a shark?" Usually there's a mix of both.
Like other comments here and on the OP have said, Deep Springs has many different cultural moments because of the short turn over of the students in their elected positions (and as residents), the 2 year or so retention rate for staff, and new visiting professors 4 times a year.
To your question of scale -- the college is actively against scaling. The founder hoped the school would inspire similar schools. And over the last decade or so, Outer Coast College, Tidelines Institute, Thoreau College, and Gull Island Project have all started programs based on Deep Springs. They're all coming out to the college next weekend for a summit.
Because students are constantly balancing too-much work between academics, labor, or self-governance, each year, the application committee finds a new stable point of work load each year. Based on my experience, if the students only had to work on the application committee, their first impulse would be to spend more time on it. The stakes are higher for them than it is for us, since, as the OP notes, the student body retains the authority to regulate the conduct of its members.
The academic program is relatively conventional, i.e. 4-10 person seminars with most students taking 2-4 per semester or term. The difference is the collective striving for great academic performance (written and spoken in seminar) and the ability for professors (long-term and visiting) to pitch courses they wouldn't be able to teach elsewhere (for any reason, e.g. politics, student quality, etc.).
These days, class is in the morning, labor in the afternoon, governance as schedules allow (though there are two regular meetings each week, committee meetings, such as applications, and the student body meeting). There is also a long-standing public speaking class each week during sept.-may; students give speeches on common prompts or speak on something important for the life of the community.