I really cannot believe, that your Pediatrician said that he is not contagious.
He is sick.
He has coughing and runny nose.
He is contagious.
The Preschool, probably does not want him there because then often times, other kids and them, get sick too, with who knows what.
Even in an office, co-workers being "sick" like that, is no fun for the others and they WISH they would just stay home.
The Preschool is probably just being cautious.
Just as a side note: so my son's friend, wanted him to come over to his house to play. And then the friend says "My Mommy is sick, she's all HOT...." So then, I ASK the Mom "Are you sick?" (and she sounded awful and looked awful when I spoke to her). And she says "yeah but I'm fine, the boys can play here..." So I say "what are your symptoms?" And she goes "Oh nothing really, just runny nose (which was obvious), sore throat, congested...." And so I say "Well sorry nothing personal but you're sick, there is a contagious period and incubation period and I don't want my son at your house and he gets sick too, and misses school next week...'
Duh.
So then she says "oh... yeah I guess I might be contagious..." meanwhile, she is having mucus fluids runny nose etc. and sounds all stuffed up.
Duh.
At least she was not offended.
And I told my son, you are not going there when the Mom is sick, and then who knows who else gets sick.
They thought, it was no big deal she is sick and inviting other kids over to play.
But it was to me.