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Lots of governments don't have a President as head of state. A lot of countries with Presidents could operate the government without a President as the true political power is held by a different official with some form of elected mandate.
The United States Federal Government absolutely needs a President to function as the President is the head of government and all governments generally a head of government to function.
No one really knows what would happen if a President isn't elected by the start of their term. Likely, the Speaker of the House would become President as they would be the first official in line for the President who held office. If the election is that far up in the airb that the executive and legislative branches aren't functioning, it could potentially go to the head of the Supreme Court, but that isn't a law.
President is the head of government
The president is the head of one of three branches of the government.
Head of Government is an established role in diplomatic speak that can be used across various types of governments to describe the person in control of what the USA would selfdescribe as the Executive Branch, but basically a government's administration. In parliamentary systems, this role is usually held by the Prime Minister or equivalent.
This is a different role from Head of State, which is typically seen as the public persona of the state in diplomacy.
Some political systems, like the USA's, combine the roles while others, like UK and China, keep them separate.
Some political systems, like the USA's, combine the roles while others, like UK... keep them separate.
... unfortunately :-(
Honestly, I think a major problem during Brexit was that the Prime Minister was relying too much on monarchial powers to push Brexit through.