If I can ask, Thande, what was Jared's logic? What's the point in an arbitrary imposition of an arbitrary monarch on an arbitrary choice of country? I can't see what they would think it would achieve.
Because the Swiss Confederacy was a republican state and the Congress Powers were a tad paranoid after the French Revolutionary and Napoleonic Wars. Also because Prussia and Austria had more influence at the Congress and Britain had less, because in Decades of Darkness Napoleon defeated Wellington at the Battle of Waterloo only to then be defeated in turn by Blücher in a separate battle.
This is part of the story, but there was more going on.
Austria and Prussia had greater influence than in OTL due to the circumstances of the *Battle of Waterloo. However, Britain was also more anti-republican than in OTL, thanks to worse experiences of the *USA in this TL, and so was more inclined to go along with the anti-republican sentiments of Austria and Prussia (and, for that matter, Russia).
The specific proposal to instal a monarchy in Switzerland had other reasons, though. Prussia (thanks to its greater influence) was pushing to take all of Saxony, in exchange for cessions elsewhere. (Less of the Rhineland than in OTL, for instance). Given the principles of the Congress, though, they weren't going to do that unless they could find a suitable new throne for the displaced King of Saxony.
Switzerland was the place chosen, but the deal was more nuanced than simply imposing a random monarch on Switzerland. The deal was that Switzerland was that Prussia abandoned its claims to Neuchatel, giving that to Switzerland as a new canton, in exchange for accepting the King of Saxony as a monarch.
In OTL, Neuchatel became both a Swiss canton
and a Prussian principality in 1815, which caused problems later. ITTL, Switzerland gets Neuchatel and a monarch whose declared role is to arbitrate between the cantons (something which did cause some problems in OTL around this period). Switzerland isn't entirely happy with this deal, to put it mildly, but the Great Powers in Vienna are all lined up in support of it, and the gain of Neuchatel does make the deal at least somewhat palatable.
(The monarchy would eventually collapse, but that's another story.)