No, the President of United Earth (as seen in DSC) is Earth’s head of state.In the 22nd century, Earth seemed to have a Parliamentary system (they had "Ministers" and what not). I think it's entirely possible that in becoming the capital world of the Federation, Earth integrated that into Parliament... like the President of Earth is the Head of Government, but the Federation President is the Head of State.
The Prime Minister of United Earth -a position implied by the existence of Ministers - is the head of government.
That is what is known as a parliamentary republic.
The President of the Federation is only the head of state (and of government) OF the Federation. He has nothing to do with Earth’s local government. That is an entity unto itself.
Remember, United Earth is just another Federation member world, It enjoys no special status or privileges. True, the main Federation government buildings - and Starfleet Command - are indeed located on Earth, but that doesn’t mean anything. They’ve got to be somewhere, amirite?
Take Ottawa, for example. It is the capital of Canada, and Parliament Hill is located there, but it is still a normal Canadian city just like any other.
Same story here. Jaresh-Inyo, for example, has his offices on Earth. And during the Changeling crisis, he has the right - as Federation President - to declare martial law on ANY member world, and to ‘federalize’ that world’s local forces as he sees fit. But that still doesn’t mean he has anything to do with day-to-day local government. He’s just exercising his authority, as UFP President, in a crisis situation.
Edit: Paging @Sci! Paging @Sci! I summon thee!

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