^Sure the President has to win some sort of election but is it by popular vote of the electorate or does the council elect the President from their members?
I'll never understand why this is such a popular hypothesis when it stands in defiance of basic nomenclature. If he were elected from among the Federation Council by the Council, then why wouldn't he be called the Federation Prime Minister? There's only one presidency in the world where the President is elected by the legislature from among the legislators -- the South African Presidency.... which evolved from the old apartheid-era South African Prime Ministry. The clear outlier.
Every other state that has the head of government elected by and from among the legislature calls that head of government "Prime Minister" or something that translates to such. Hypothesizing that the Federation President is elected by and from among the Council adds another layer of needless complication with no evidence to support it.
Occum's Razor: The Federation President is an actual president, not a prime minister, and is elected by the people.
Actually, the President of the United States is not elected by the people either. When the people vote for the president they are actually selecting the slate of electors who do the actual voting on December 20th of an election year. Federal or a State's Congress tomorrow could remove people from voting on the President and it wouldn't change anything. I can say this because voting for President by the electors is mandated in the Constitution of the United States, while the popular vote isn't. Each state determined its own preferred method of choosing the electors. There were some states where the legislature chose the electors and in others it was done by popular vote. It wasn't until the late 1890s til the early 1900s that all states used the popular vote to choose the electoral slate.