I never had this issue come up because I have 1 child, a daughter age 20.
I believe what you mention is a good idea for small children but in no way would I force my son or daughter to be in the room when the other was having an exam at the ages your children are at this time.
I assume you want your children to have a relationship with the Dr. Well, I can tell you that a 10 yr old girl approaching puberty will not be open with the Dr, ask or answer questions. Same for the 9 yr old boy.
Each child is old enough to know what goes on at a check up and deserves respect and privacy so they can answer and ask important with the Dr.
They are old enough to sit in a waiting area but if you are so against that, at least have them wait outside the room. Please give them privacy and respect that, especially at these ages.
I stopped going into the room with my daughter years ago. She is old enough to speak with the Dr. in private and ask anything she wants. If the Dr. needs to see me, then when the visit is over, we all sit together and discuss anything to be discussed.
Your children are growing up and they need to develop their own relationship with their Dr. privately and respectfully.