Sounds like a good opportunity to educate your co-workers and the child care center about how to communicate the seriousness of this issue. The person answering the phone could have simply thought your child was not very ill and the child care center could have insisted on speaking to you directly.
But! You should always have other people listed on your pick up card at the child care center so that when something is going on and you and your husband cannot be reached where someone else has the authority to pick up your children and take them to the doc.
You can sign a paper at the docs office giving permission to treat and a list of names or a paper for each individual. You can also contact your local hospital and go in and sign an affidavit or other notarized document stating that "so and so" has legal permission to bring your children in for emergency treatment.
This way if you are on a field trip and your husband has something happen where he can't get to the school soon enough, maybe he goes out to the car and has a flat, or has an accident on the way to the school, or vice versa, etc... that the children can still get the medical treatment they need.
And in my opinion:
If the child care center knew this was a serious issue they should have insisted on talking to someone else that had the authority to allow the phone call to be put through or to relieve you from your class duties. A secretary really has the job of fielding phone calls. It is their job to take messages for how ever many teachers there are. It is not the secretaries business to know your child has any specific illness issues. Privacy is important to you too.
If the child care center had said it was an emergency in the first place then the secretary would have informed the Principle there was an emergency phone call from your child care center and the proper procedure would have occurred.
Child care centers call for parents to come pick up children all the time and sometimes it seems a very unimportant thing. If a parent can't come pick up a child right away we just separate the sick child and keep them comfortable until someone on the pick up list can be reached and come get them.
Just seems like an opportunity to clear the air with everyone.