I would definitely be excited about a school that offers many academic extra activities. To me it speaks of the focus of the school--the importance of academics--and rewards for high performance (that's the nature of those competitions). Plus it says staff are devoted to sponsor these things after hours. The administration supports it--they've authorized funding for the competitions, buses, maybe coaching stipends. It says a lot of a school.
I would think your son should be able to participate in many of these, if not all, depending on how it's organized. Chess club may be just an afternoon every couple weeks (or it could entail tournaments). Science Olympiad is November-February and can probably be 2x/wk. Spelling bee is a one time thing unless your son wins district, then he'd go on to regionals, then National. Clearly if he were preparing for National Spelling Bee it would require giving up most other things. Math olympiad should be once a week, one hr, probably November-March.
The problem I see is how well organized/coached are the clubs/competitions. I often find that I have to throw myself into the activities that my kids are doing to 1) drive them there and back 2) help them get organized (a lot of time a coach will ignore the really smart kids and help others, but being smart doesn't mean you know how to organize or have direction in something you've never done before 3) a compulsion to make it better/more meaningful. If you have to throw yourself into these activities, then you might have a limit before your son does.
