Seems we're on the same wavelength because I had the exact same thought about this.

But actually, I'm probably not so fussy. I wouldn't mind cleaning up the mess if I'm paid hourly... But now my concern is: What if they want me to clean up the mess but demand that I do it in a particular way which makes it impossible?

I'm already seeing signs of this. This was already kind of the case in my last job; I had to fix things and implement new features but we had to keep the same clunky, over-engineered architecture. My current job doesn't have the same degree of architectural legacy baggage but there is a lot of bureaucracy to deal with instead, which is itself restrictive.