I think it is unreasonable to turn a child away for a cold.

You would need to have that VERY clearly written in your contract and agreed upon before turning them away.

It sucks to be constantly exposed to illnesses but it does come with the job. We do our best to prevent spread and we exclude for the big contageons (as you have noted) but colds are a normal part of childhood in group care/school etc.

I would suggest doing everything you can to boost ypur and your families immune system to help you fight off the colds. Paying attention to sleep, diet, lots of water intake etc. Every bit helps.