Someone might expand on this question, but since it has gone unanswered for so long I think a brief response might also be considered useful.

First, the 'old republican system' is not quite so simple a situation. Prior to 1813, the area known as the Netherlands had been ruled by the French Empire of Emperor Napoleon (1810-1813), the Kingdom of Holland of King Louis Bonaparte (1806-1810), the Batavian Republic (with various coup d'etats, 1795-1806), a period where stadtholder William V reigned with the explicit backing of his Prussian brother-in-law (1787-1795) and then a turbulent period in the 1780s when his rule was challenged by the Patriotten (patriots). One might even go all the way back to 1747, the year that saw William IV become stadtholder and secure for his family (a different branch of the House of Nassau from those who had earlier held the title of Prince of Orange) an inherited position as stadtholder. That was new, as stadtholders had previously been chosen by each of the individual States.

So if one were to have said in 1813, 'let's just go back'. The obvious problem would have been: to where?

The shadow of William of Orange has always loomed over Dutch politics, and as such the Republic existed for many decades with various Princes of Orange as the leader of the army and navy, and effective head of state. Although these periods were sometimes interrupted (after the death of William II, from 1650-1672 and after the death of his son, and last in that original line, from 1702-1747), the princes of Orange were never far from the scene of the action in the Republic. To briefly go back to that earlier point: William IV was of the Frisian branch of the Nassau family; and this 'Stadtholderless period' refers more specifically to Holland. Friesland did have a stadtholder, just a different one - and one that Holland refused to appoint.

Back to in 1813. Three noblemen took control of the government as the French Empire crumbled, and they invited William V's son, William Frederik to be 'sovereign prince'. This wasn't an explicit promotion to kingship. Indeed, earlier princes of Orange had been referred to as 'His Highness' and had always combined their princely title with the Republic. The entity that came into being in 1813 was thus the Sovereign [i.e. independent] Principality [i.e. headed by the Prince of Orange] of the United Netherlands.

It is the Congress of Vienna, led by the monarchies of Europe, that eventually gave the green light to the creation of a Kingdom of the United Netherlands. But before that treaty was ratified, Napoleon returns. It is in the chaos of that return that William Frederik seizes the moment and declares himself king of the United Netherlands. Unfortunately for William Frederik, the southern Netherlands do not consent to the draft constitution. That is, until William Frederik 'reinterpreted' the results of their votes (mockingly dubbed arithmétique hollandaise in said southern provinces) and declared the constitution adopted; and with it, the kingdom created. With the consent of the kings of Europe, that is. The period around 1815 is not a time particularly keen on the (re)creation of republics, as one might imagine.