Switzerland doesn’t have a king because it’s a federal republic, not a monarchy. Further, Switzerland’s history as a confederacy of independent states never lent itself to the development of a monarchy or king.
Why Doesn’t Switzerland Have A King?:
Switzerland has always been a very decentralized country. It began as a small confederation of semi-independent states that was born from the desire to be free of overlords like the Habsburgs.
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At the time, the House of Habsburg were trying to gain control over large areas of Switzerland. And the first members of the fledgling Switzerland at that time were in no mood to become subjects.
On top of this, Switzerland had a history of being ruled by the Roman Empire in the past, which no doubt fueled this lust for independence.
Unlike many other European countries, Switzerland did not have powerful noble families with dynastic claims that could push for the establishment of a monarchy. So, over time more members joined the loose confederatin until Switzerland gained independence from the Holy Roman Empire in 1499.
So, rather than having a history of ruling parties and fights for power, Switzerland has one that was dominated by foreign rulers which leads to a fierce drive for independence that still persists today.