Mainly I agree with your points. Here goes...
a) School leadership requires both domain-specific and domain-general knowledge/skills.
BUT...
c) We should try and re-balance this without over-reaching/claiming the argument.
'Organizations perform more effectively when led by individuals who have a deep understanding of the core business of their organization. Being a capable general manager is not sufficient.'
(1) inherent knowledge, acquired through technical expertise combined with high ability in the core-business activity;
(2) industry experience, which stems from time and practice within the core-business industry;
Full paper here: papers.ssrn.com/sol3/papers.cf…