100% keep your logbook yourself. It is YOUR logbook. There is no valid reason to leave it at your flying school.
Random CASA inspection? BS. Yes, CASA may ask and arrange with you to see it.
Mrs 440 is in a local community organisation, they met weekly at a local pub, left all there stuff there for convenience. The pub went broke, closed, locked up, owners could not be found. By the time the group was able to get back in there, the whole joint was cleaned out by a contractor presumably, everything was just gone, including all the groups property which was clearly marked with their well known branding.
Couldn't happen to a flying school? Dream on. Keep your logbook yourself.