I would suggest that you get EHIC cards ( the new name for E111). They are free so why even debate it.
If you have an emergency and an ambulance is called, the local public ambulance personnel may not be interested in your private insurance and may only be in interested in whether you are entitled to be treated in the local public hospital, which the EHIC proves.
What does your travel insurance say about what to in an emergency ? Are you advised to call an international number to find out about what local private services you are covered for. This is probably different per country, but you really dont want to have to do this when you need emergency treatment. What private services
exist in the country you are visiting ?
You dont say what country you are visiting, but in France, for instance, the public health system is brilliant, it is where you want to be if something happens.
Private health insurance is another backup, if you needed an air ambulance home or
further treatment but may not be your best option in an emergency.