Maybe! Depends on the village counselor. All I know is after quarantine you have to register your negative vaccine results with him. So might be worth having because he has the right to isolate you should he decide.
I don't know about phuket sandbox I am in quarantine in Bangkok maybe it's different? Also this was the application procedure from the UK. Im sure you are right about what you say. We have just been through different experiences. Good luck enjoy your holiday!
just a quick warning Peter. I see this question is getting many replies. Don't be confused. You need a document about where you go after quarantine hotel. I would stick with your original plans. Thai immigration are not stupid. If you don't have a onward flight they are less likely to require a return ticket. Also they are aware of people arriving hoping to get a covid extension. It's by no means guaranteed. I'm in quarantine right now. Good luck!