I feel two ways in this, you probably need one voice for certain key tactical decisions, but the fact that everyone feels the presidency is this important in America supports the idea that the presidency has gotten way to powerful. Loosely speaking, the person who is president shouldn’t matter as much as who is in congress.
This is how it works in most European democracies.
The balance between the President, the Prime Minister and the Congress or Parliament is different in different countries. In most of Europe, the President is mostly a "figurehead". The President does the diplomatic stuff, wines and dines with foreign leaders, maybe has the ability to veto laws or pardon people, but it's the prime minister who actually runs the government and appoints the cabinet.
I think this varies varies between countries. Sweden indeed has an almost (formally) powerless monarch. I think the UK has a monarch with some real power but that does not put it to use? And how Denmark, Netherlands, Norway etc fare on this scale would be interesting to learn.
> you probably need one voice for certain key tactical decisions
I agree. Having a unified vision is the other key thing a president brings. Unfortunately though we barely see the fruits of a unified vision with how divided the two parties are today.
