I think Vortigern99 and DragonLady have established that the costs to the zero-tolerance approach to the students are significant, and the cost of having the school nurse manage the meds for potentially thousands of students is also going to be non-trivial.
I would disagree, as this is the policy for all the schools in my current district, and in most others my children have atteneded. Seems fairly common. If it were cost-prohibative, I would expect that to be reflected in current budgets where this is the policy. Add to this that Midol (just to take the OP example) offers an extended capsule good for 12 hours...can be given at home and no need to involve the school at all. This is true for many medication with 8-hour or longer times, which covers quite a bit.
That's apart from the previously mentioned cost of at least one child dying because their asthma inhaler was in the nurse's office not in the child's pocket. That's a cost too.
And the school policy (again, at least the ones I am familiar with) make a specific exception for medications of this sort to be kept on the student's person, usually needing a copy of the prescription and a note from the parent. I would be suprised if this isn't pretty much standard at public schools. It wouldn't suprise me to know that a similar policy existed, even at that location, and the parents simply never read the manual to know the policy or followed the steps required to allow the child to have the inhaler on his person. That being said, without knowing the details, that's just speculation.
I'm not at all clear what you think the benefits are that outweigh these costs.
The benefits are mainly in the schools capacity as surrogates for the parents. A lot of the policy is to make sure that:
The parent knows the child is taking medication
The school knows the child is taking medication
The school nurse knows what medications the child is taking, so as to better recognize drug reactions, and avoid giving contra-indicated drugs or treatment to the child if something else goes wrong during the day
The first one is a key element, I think. Do you allow your children to simply buy whatever OTC mediciation they want and take it as they will? If your children are older and you trust them, that may be. But for some, and especially for younger children, I would not make that decision for someone else's child. And that's basically the school's position.
As stated before, the risks are things like potential drug interactions, poor drug choices (OTCs taken when the student has a contra-indication, such as NSAIDs with gastric problems or Aspirin with flu), lack of control, a lack of awareness of the school that a child is sick/having issues, and the potential for the spread of illness among the student population. All of these are directly or indirectly addresses by a policy of having the nurse involved.
Frankly, I'm amazed at how most adults use OTC medications, and if any adults were under my care for any reason, I would require that they notify me of any medications they were taking as well, whether OTC or not, as well as herbal suppliments and dietary suppliments (which can also cause various non-trivial interactions with some OTC and prescription medications). The children are the responsibility of the school, so the school needs to (at the least) be aware of the child's conditions.
Again, that being said, I think a manadatory suspension for any violation of the policy is rather stupid, but I don't have any issue with the policy itself (the policy of no drugs except thorugh the nurses office or with a prescription for emergency medications). I don't think zero tolerance, in the sense that everyone who violates this gets the same punishment, makes any rational sense at all. It removes all element of proportion when punishments for taking an OTC yourself and snorting a line in the bathroom are essentially punished the same, and I don't think it sends the message the school intends. I don't like zero tolerance, but I do see the point of having a policy against even OTC drugs. I would be alright with an exception to policy for certain common OTC meds, where the child can keep and self-administer with a note from the parents. But I think a policy of "anything goes" is about as irrational as zero tolerance is.