Hi there, a fellow chief of staff, an Insead alum and admissions consultant here.
It's absolutely fine applying to Insead with a 9 month work-ex as long as you are able to show a line of sight for the projects that you have come onboard for (from now until Jan 24)
For example, a recent client of mine was hired to start a new biz for a startup and scale it to certain goals over 15 months. And he applied to Insead 2 months into his job to secure admission. Personally, I'd applied 3 months into my new job when I got selected to insead too.
The things that I'd be mindful about are -
A) Whats your personal learning curve going to be like in the role. CoS roles can be incredibly rewarding as learning experiences. Hence soak it in, and if that means applying to the next cohort, so be it
B) Focus on your narrative of transitioning from a core biz career to a startup and connect your current learnings to your future goals
C) Hope you have had global exposure through your past roles
Good luck. Happy to chat more should you want to.