“We can’t have a Jewish president.”
Not that we can’t, as in it isn’t possible. Of course we can. We (or you Americans) should not have a Jewish President. It would mean disaster for Israel-US relations as s/he would over compensate to the Arabs etc to show s/he is not showing favouritism to Israel.
I can’t comment on US history, but Tony Blair’s government was the most pro-Israel in recent times, as there were no Jews at all in the cabinet.