From what I can tell, every state (all 50) has a school vaccination law that lists required vaccinations and has an additional provision for the state health agency (an executive branch agency under the governor) to add additional requirements as warranted. I think you can probably do the legislative history on those and find that the flexibility to add new vaccination requirements without the need for new legislation, as long as it meets health agency safeguards, is sensible - particularly in a public health crisis situation.
California's school vaccination law allows additional vaccinations to be added, "taking into consideration the recommendations of the Advisory Committee on Immunization Practices of the United States Department of Health and Human Services, the American Academy of Pediatrics, and the American Academy of Family Physicians.
I just looked, all three of those organizations recommend Covid vaccination for school children 12 and over in accordance with current authorizations.
Where's the "too much executive power" part?