As you noted, generally (with specified exceptions), the professions for professional corporations are required (for which LLCs are prohibited) are those for which services are rendered pursuant to a license, certification or registration authorized by the Business and Professions Code. (Please see the Corporations Code Sections at the first link, below.) Child care providers are not included in that set of professions.
Dealing with your situation more directly, Health and Safety Code Section 1596.80 (second link below) states that and "person" maintaining a child day care facility in CA must have a current valid license. Furthermore, Section 1596.79 (third link below) states that the definition of "person" includes a limited liability company. So a preschool may be operated by an LLC.
This information does not constitute legal advice and does not establish an attorney-client relationship.