The new system is just like the old. You only need to produce the card if and when you need medical treatment. No need to register it any where else.
As a reminder the EHIC card is to get reciprocal treatment in another European country, ie, like the NHS, going to your GP, minor A&E as you would at home.
Travel Insurance is for major items on the medical side, such as paying for hospitalisation, geting medivac back to the UK, etc etc.
If one did not have the EHIC, you will have to pay, and then claim back off your insurance a later date.