Yes.
Here's a possible scenario:
Your daughter is unfortunate enough to have an accident which requires hospital treatment. As a non-national, she's presented with a bill which must be paid before she's allowed to leave the hospital premises. If she shows her EHIC, she'll be entitled to the same free service that local residents receive and the bill will be waived. If she doesn't have it with her, she'll have to use all of her holiday cash (and possibly much more as well) to pay the bill and then try to recover it from her insurers. (Some, possibly most, insurance policies won't pay out in situations where the traveller wouldn't have received a bill if they'd carried an EHIC, so there's a possibility that she might be left out of pocket).
When travelling in countries where the EHIC is valid, it's important to carry it at all times. It ensures that the traveller will pay the same medical fees as local residents. (In most, but not all, cases that means free treatment). The medical element of travel insurance is really only required to cover expenses not met by the EHIC, such as the cost of an air ambulance to bring a severely injured traveller home.
I often travel in Europe without travel insurance but my EHIC resides permanently in my wallet. (I'm far more likely to, say, break my leg than to require an air ambulance home).
Chris