Yeah well, I already started the run as a Jurchen tribe to get as many achievements as possible at onceI don't think you can at take EoC if you want the achievement. You could start as Oirat and form Yuan instead because that doesn't come with forced government change provided you completely annihilate Ming (or any current EoC).
Got the Pyramid of skulls one, the other ones I´m aiming for are
I´ll graze my horse
A Manchurian Candidate
Blood for the Sky God
Qing of China
These Banners need a Saga
and Turning the Tide.
So I have to get EoC for Qing and Manchurian Candidate, but I guess I could keep Ming as a isolated nation inside my borders and prevent anyone from taking the mandate from them so it doesnt stop existing. There is no crisis or event that deletes EoC, right?