Good grief, nothing wrong with next of kin and certainly not morbid! It is a sensible idea.
If you are experiencing a medical emergency, you won't really care what term is used.
As someone, who has experienced a number of life threatening medical episodes, I am well aware how important it is for the need for someone, who can be contacted.
My medical POA is recorded on my NHS records, that way my next of kin can authorise treatment and be included in decision making , if I am unable to do so. I can confirm that it has happened on a number of occasions.
As for accessing your contact details, by knowing your name, it is not always the case.
Last year I was admitted suddenly to a major hospital with a heart issue. I bypassed A&E and was taken straight to the cardiology unit.
Despite attending numerous clinics in that trust and records computerised, they could not find my records.
I keep my CHI number and an emergency treatment card in my purse.
Bingo!