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.