Sooooo I finally figured it out (to some extent). I'm not saying this is the right way to go, but it solves the issue. What I did was to put the following into the XamlSharp.config (a config used by Sitecore.Kernel):
Inside the tag I added three tags:
<htmlControl tag="TextBox" type="System.Web.UI.WebControls.TextBox, System.Web"/>
<htmlControl tag="DropDownList" type="System.Web.UI.WebControls.DropDownList, System.Web"/>
<htmlControl tag="CheckBox" type="System.Web.UI.WebControls.CheckBox, System.Web"/>
Making the entire look like this:
So, how did I figure this out?
I made a pdb of the Sitecore.Kernel.dll and debugged my way into where the error was raised:
Since the controlType nor the subclass of the controltype didn't match any of the htmlControls, the error would be raised here. But why aren't the webControls in the list of these controls? And why does it work on other instances of Sitecore 8.2 update 3 (I tried at some colleagues environment, with a different codebase)? I have literally no idea, and since I'm out of time, I will have to stop my investigations at this point. :(