Some schools enter one off teams in carnivals, and some of the State Schools like Epping BHS, Homebush BHS play semi regular (more recently these have become more in the way of irregular) Wednesday games against each other, but outside the AAGPS (9 schools), CAS (6 schools), and ISA (about 15 schools across 2 divisions) there is little school rugby action in Sydney and surrounds.
Junior Club rugby is the only real option. In saying that, despite the grumblings on here about SJRU from time to time, your son (and plenty of others before him) can have a great time playing junior village club rugby right until he leaves school, after which Colts in Premier Clubs, or Suburban Rugby Clubs beckons. Shute Shield clubs are getting much better at engaging with their Junior Clubs and the SJRU pathway.
In the older age groups, there is a bit of a skills gap between what the Big Schools offer, but until Under 16's some (if not all) schools turn a bit of a blind eye on their kids playing Club footy on Sundays, (including those in the Age Group A teams. It gets quite serious for the 16A's (sometimes 15A's) and the first XV, second XV (and sometimes third XV), with many schools actively preventing kids from playing club and school footy, mainly on player safety and recovery grounds as the physicality in the game increases exponentially once the Puberty Fairy waves the magic wand.
Plenty of young folk from the Big Rugby Schools give up the game after they leave school, and similarly plenty of professional or top level Shute Shield Club players did not attend a Private School with a big rugby programme.