No. EHIC only covers you under reciprocal agreement in the EU that entitles you to the same treatment in hospitals as locals get. If the locals get free health care, then you do too. If they pay for some of it, then so do you. Only for emergency treatment though, and only state hospitals.
It won’t give you recovery off a trail/mountain or ambulance trip if they normally charge for it, or repatriation. It doesn’t replace travel insurance cover for non-medical stuff.
You still need it though as many/most travel insurances will require it and they will provide additional cover beyond EHIC.
If you have travel insurance, check the small print. Mine says MTB but “no extreme terrain” and on enquiry they’re vague but suggest anything other than gravel or tarmac is excluded.
Cover for theft, check your house insurance. Some might cover it, even overseas.
If you want serious insurance, go for Snowcard, Dogtag etc, but sounds like it would be OTT for this kind of trip.