There are plenty of non-scholars in HQ. Since entry level position is SEO1 level though your CEP probably needs to have been increased at least once (even if you've not taken up the role). If you're already holding an appointment, should qualify. Depending on your relationship, could possibly let your RO know you are interested in applying and see if he/she dissuades you.

I wouldn't say they deliberately groom young officers, but being used to taking in new officers who get posted back to school every 2-3 years I would imagine most divisions have processes in place to manage the assignment of work, e.g. You might be a team member in your first year, but leading project teams by your last.

Prior to joining HQ I felt rather incredulous about all these HQ officers parachuting back to schools as middle managers, but now that I'm here I can see how the nature of work does prepare you for managerial duties (even if in a somewhat different context).