DMB - one thing that comes to mind is vaccinations, or other government health programs. If these are considered medical treatments by the medical provider that is consulted by the educator, then these could be theoretically covered by the statement.

Certainly, administering a tetanus vaccination in the case of many injuries would be highly probable by a treating physician. However, they would need your permission for this as far as I am aware, and the centre have access to your immunisation records.
I think it is a valid concern you have, but not one your centre can answer for you, nor one for which they are responsible. Perhaps it is worth contacting the AMA and asking for emergency treatment of minors guidelines or pertinent legislation?