“Many countries have had Jewish heads of government or heads of state, and in most cases there were no issues. Leon Blum may have been an exception; he was a Jewish socialist and the anti-Semites of the political Right preferred Nazi rule to the Blum’s Popular Front. But Blum was not religious. “
While many of the aforementioned were possibly Jewish and likely aligned with the OO mentality, the OP’s question was about Halachic issues for a Frum president.