Circle leaders don't burn out, the kids see different approaches to math, and the leaders only need to prepare a few sessions that can be repeated at multiple circles. Circle leaders include teachers, professors, graduate or even undergraduate students, and other professional mathematicians. Circle sessions are focussed on a particular topic. There is homework, but exciting and seductive homework. Math circles prepare students for Olympiad-style problems like those of BAMO, the Bay Area Mathematical Olympiad.
Circles teach kids to be mathematicians who solve essay-style problems requiring proof.
Hand out a set of problems a week before your session. Not too many, but seductive. Include an easy one and a challenging one. Try not to lecture. Even though introducing new theory and techniques is an integral part of math circles, your sessions should be as interactive as possible.