I think you brush over a detail too fast. The US president needs to unite his party… until the last ballot is cast. That very instant, this stops being true for four years. Combined with a powerful executive that keeps the president very powerful even without legislative support.

Of course by definition any democratic system has checks and balances and ultimately ends up being representative of the will people in some way, but my point is that British PMs are a lot closer to being “harmless distractions” such as Zaphod than US presidents (also Douglas Adams was English).