The short answer is yes and no.
There are many symptoms that a child may not attend child care with, from something seemingly minor like a new rash, to something more complex and potentially serious like a recent head injury. Children need to be healthy, well rested and able to keep up with their peers and the schedule. The needs of the group for supervision and safety also need to be taken into account; if an educator is spending their time wiping a nose that's constantly running, and then washing hands, supervision of the group at large could be compromised to a degree. Not only that, but these things can spread like wildfire amongst a group too young to properly cover coughs and sneezes, and adequately wash their hands. However, mild cold symptoms are generally not a problem, so long as children are still happy, playing, eating and resting as normal. The brass tacks of it is, if your child is too unwell to participate fully and happily in the day home routine, they need to be at home to rest and recover.
The goal is to keep everyone as healthy as possible within the program. It is inconvenient to have to take time from work when your child is unwell. I have endless empathy for this situation. It must be made known though, that exclusionary symptoms for children in care also require that I close the program if I or a family member who lives here become afflicted. This shuts down everyone's access to child care. Therefore, keeping your child home for a day or 2 to prevent spread at the onset of illness will actually aid in securing your childcare in the near future, by preventing me and others from becoming ill and having to close temporarily as well.