Of course, one question that occurs to me is "why wouldn't they simply not nominate a stadtholder after William dies?"
It's not a hereditary title, after all.
Actually I think that is very likely to happen. Maybe not if William dies leaving an heir. William had to much influence for that to happen, assuming of course the heir was old enough to succeed him. If not you could get the same situation after stadholder Willem II died. No stadholder elected.
My guess is that the following would have happened:
William III dies leaving a som (Willem/William) old enough to succeed him both as stadholder and as king of England, Scotland, etc. Holland, Zeeland, Utrecht, Gelderland and Overijssel appoint him stadholder of their provinces (leaving Groningen and Friesland to Johan Friso). Willem IV proves to be more William IV than Willem IV, he pays far more attention to England than he does to the Netherlands. When he dies the Dutch provinces decide they rather have a Dutch stadholder than an English one and appoint not the first, but second son of William IV as their stadholder or possibly appoint the stadholder of Friesland and Groningen as their stadholder, ending the personal union.
Eventhough this might be my opinion of what would have happened, Flocc's timeline is still a great read.