You can mention whatever Executor you choose in your Advanced Directive, a copy of which the doctors ask me for every time I visit. Even if there is not a section for it, you can add pages to any ADR. I gave a copy of mine to my kids. Why does the hospital want one also? In case the family forges documents, I guess, right?
But that is one way to do it - give a copy of your ADR to your PCP and the hospitals you usually go to. If you die at home, have your Executor's contact info plainly laid out on a table or on the frig. Better to have it in at least three obvious places at eye level at home. In your hospital chart, IF anyone is paying attention to that, it should be in a special section for the ADR.
Another thing you need to think of is whether or not you want or need a Medical Power of Attorney. What's that for? Well, I've had people in my family at BAD hospitals as they were very sick and I could not get them moved to a better hospital because I did not have a Medical POA for them. With one relative I told him he should have one, we should ALL have one, but he didn't listen to me.
You can be so very sick that you cannot speak. Who will speak for you then? That's what I need to get next - a Medical POA so if paramedics take me to a Hospital From Hell, my kid can walk in with that POA and say, "Move her or I will sue you STAT." I should pay a lawyer to prepare that so my kids can call the lawyer if Beelzebub's Hospital, Inc. wants to argue with my kids, who would be upset.
We have a few of those in my area - Beelzebub's Hospitals, Inc. But, that is inevitable with Medical Care, Inc.