K.W.
I knew someone who watched a child go into anaphylactic shock from the smell of the peanuts at the ballpark. So, no it's not overkill. The really scary thing about nuts is that they are oily. So you can think you've cleaned off a surface, but it's still contaminated.
I hear where you're coming from. I do have a suggestion, but it would be a lot of work. If you have the time, energy, and inclination, you might ask the teacher if you can take over the task of providing and coordinating snacks. If the teacher approves, maybe you can send out a letter something like this:
"Let's give our kids a year of the best snacks ever! I am the new volunteer snack coordinator, and I have a passion for feeding our kids well. If your child has any special dietary needs, please contact me ASAP so we can take that into account when menu planning. Otherwise, please pick one of these options:
1. I want to help prepare terrific snacks for our kids.
2. I want to give a cash donation to help buy terrific snacks for our kids."
I strongly suspect that, for the most part, the parents bringing in pre-packaged, processed, unhealthy snacks would be truly delighted to donate money instead of a box of overpriced crackers.
If you will be preparing things like fresh fruit and vegetables, it wouldn't hurt to go get yourself a food-handler's permit. It's not necessary, but they're readily available, the test is easy, the permit is cheap, and it will provide some peace of mind for all parties.
Good luck!