Can EHIC be such document? :D
Yes, it can. It's what they normally advise for people who have just moved to Poland.
Poland wants to see that you are insured on a full-time basis in Poland whilst you are in Poland, for this you need evidence of private medical insurance
It's not needed. It's enough to pay ZUS as an employee of some sort (or be self employed), or to use the EHIC. Or you can be covered through the NFZ through a family member in the same household that pays ZUS contributions, too.
Poland is rather laid back in this respect.