I don't think that's likely. If the President was chosen by the Council, then he wouldn't really be the President of the United Federation of Planets; in English, the term for a head of government chosen by the legislature is "Prime Minister."
This isn't entirely correct, though. You forgot countries like Germany and Italy (Germany especially, since it is also a federal republic). Both have a head of state called a President but he isn't elected by the people. He is elected by a special body consisting of Members of Parliament and state/regional representatives. Of course, the catch is these Presidents are largely ceremonial roles with little real power (though I believe they are technically still CinCs) and most of the power is with their Prime Ministers. The Federation presidents we've seen seemed to have larger powers than just ceremonial and we've never heard of an UFP PM, so I doubt it's this system they use. OTOH, I doubt the UFP Prez is as powerfull as the US one. The Federation seems to me a much more loose federation than the USA (and it should be, since it isn't truly 'one nation' like the US, but more a 'union of nations' like the EU) and I doubt one man would be given such power. Maybe they have something like France or Russia? A rather powerfull and dominant President but still with a separate PM and his Cabinet?
(and, after I wrote all this, I just realized you said head of government, not head of state...
Given Star Trek's commitment to American liberalism and the reflection of those values in its depiction of the Federation, I can't buy the idea that the Federation allows Member States to join that are not themselves liberal democracies.
The Federation doesn't seem to be completely strict though, allowing for some cultural peculiarities. Take Bajoran politics, for example. They were a democracy with a secular government, but the religious authorities like the Kai and Vedek Assembly also held an important political position (wasn't it a Kai that led the peace talks with Cardassia?). Yet, the UFP is ready to admit Bajor in.