OK...here's my 2 cents :O) I am a mom, not a provider.
I think it's perfectly acceptable to ask for a Dr's note in this situation. However, the runny nose thing is soooooo common, even for kids who are not that sick, so you might have to re-think that part of your policy.
I believe that infection lies with a fever. So perhaps you should update your policy noting that kids cannot come to daycare with a fever. And if a kid HAS been out for a few days with fever/cold, then they cannot return until they have been fever-free for 24hrs. This is a perfect time of year to do this because winter is almost here. "In an effort to try to keep our children healthy this winter, I am revising my health policy effective immediately..."
Yes, boogers and snot are gross.....especially from someone else's child :o) I can appreciate that!!!! I know most daycare's have even the simplest "routine's" throughout the day. Perhaps you could work in specific times to clean hands (even hand sanitizer)......after morning play....before we eat.....whatever.....this could eliminate some stress about all children's winter germs. At the same time to clean nose's.
The little guy in question is only 2........but even at 2 he can be taught to "cover his mouth" and wipe his nose. He should always have a kleenex in his little pocket........he may not cover and wipe all the time, but you can be a teacher to him to help him keep his germs to himself :O)
J., even with the most perfect policies in place, there will always be someone who tries to break your rules........parents are busy, and rely on daycare to help them........we parents don't always think about everyone else's children when WE need to get something done.........I mean, I do, but majority do not.....
You can only do the best you can do :o)
I hope something I've said can help you in some small way. This is really a tough one with our winter approaching. Kids are sick all the time, even when they are not in daycare. And I'm sure you're just trying to control your environment so when you are NOT a daycare.......you are a mommy with a clean house :O)
Good Luck!
~N. :o)