Was it already fully set up like that, or could a more active President have the ability to change it?
It was set up like that from the get-go. The presidency was set in the Transition Law of 5709-1949, and then defined more in-depth with the Basic Law: The Government 1964, but there wasn't really a way for a more active president to change it.
Since I really doubt Einstein had any knowledge of Hebrew, and given the recent influx of mostly Yiddish speakers into Israel, there is a possibility that he would carry on communicating in German. Including public speeches etc. Thus Yiddish would remain/become common language of Israel, with Hebrew having more solemn, official, but only de jure position (cf. modern Irish vs. English).
Though the possibilities are very slim, it is already too late I guess. But one never knows.
Hebrew had already been the unifying language of the Jews in Eretz-Yisrael/Palestine since the early 1900s. And Israel made Hebrew a co-official language of the State along with Arabic in 1948 with the Law and Administration Ordinance 5748-1948. Mizrahi Jews, Yemenite Jews, Ashkenazi, Sephardi, Ethiopian Jews all adopted Hebrew because otherwise they wouldn't be able to understand each other. And it doesn't make sense why non-Ashkenazi Jews, who have no connection to Yiddish, learn a language like that when Hebrew is already a unifying factor and there's already a solid Hebrew language learning environment?
Yiddish speakers were routinely pressured or encouraged to speak Hebrew (
ivri, daber ivrit! Hebrew [Jew], speak Hebrew!). The only people who really speak Yiddish in Israel, then and now, are the Haredi Jews and maybe a few secular Ashkenazi Jews.
Einstein would probably have a translator assigned to him, and a lot of the early State leaders would have spoken German or understood German.