I found this helpful information on the FDA website about how to tell the difference between a cold and the flu. I’m referencing this regarding one of my children who is sick with flu symptoms.
I thought perhaps this information may be of help to your family.
Note the part where this government stie says to not use antibiotics on a cold or the flu.
I also note they say not to call a doctor right away and then tell a list of when to call a doctor. I find that interesting. They also don’t say at what temperature point for the fever it is advisable to call the pediatrician for a child who is sick. I find that odd.
What to Do for Colds and Flu on the FDA site