How much time should be allocated to supervising undergraduate researchers?
One person cannot possibly supervise 200 undergraduate research projects per semester of any substance. This sounds more like a term paper with a research component than 200 fully-fledged research projects. When I did undergraduate research, it was a 20-hour per week job that I got paid for, and even then my supervisor and I rarely met for more than half an hour per week. 

There are only 168 hours total in one week, and only 40 working hours in a week. If you spent your entire job meeting with all of the students every week, you could give them 12 minutes each. This is clearly impossible. 200 students is so many that giving them each a 30-minute kickoff meeting to discuss their initial idea at the beginning of the semester would take you two and a half weeks. This would put some students at a serious disadvantage over others by potentially delaying their project start or leaving them confused about what is required of them.

If these are not much more than a term paper, then one person can probably do this by putting the assignment together on paper or in email and only answering questions during normal office hours. These kinds of assignments have clearly articulated goals and guidelines and apply techniques as laid out in lectures and homeworks. If you want a more substantial, independent project, then you need TA support, at least 6 and as many as 20 TAs to cover 200 students.