The problem is that the High Priest did things that could only be done in the Temple and the Temple could only be on the Temple Mount in Jerusalem.
This is true, and IIRC my Jewish history classes in high school is why modern Judaism is Rabbinic.
The Nasi, and I never said it was the same as the Kohen Gadol, just a good example of how somewhat similar figures operated/might have operated in the diaspora.
But I'm...not sure it is? The
Nasi's authority stems from being, in essence, the Chief Rabbi*, whereas the
Kohen Gadol's authority stems from being boss of the Temple. These...do not operate in the same way.
*Also, looking it up, the position of
Nasi was created expressly to separate the
Kohen Gadol from authority over the
Sanhedrin.