An issue with this approach is it turns 'cleaning' into a time-consuming task that needs to be completed to avoid making the situation worse. That can cause some folks anxiety and require a building up of sufficient willpower to even begin the task.

Contrast that with a transaction-based approach to cleaning where an individual item is moved to it's destination place/state, then you move on to the next item. No middle state where things are worse, and you can stop at any time yet still have gained the benefit of any work completed to that point.

However, with the second approach it's key that everything has a suitable destination 'home' to be moved to. Often, that 'home' is the trash can :)

My wife attempts to use this method, but usually gets distracted at peak-messiness and things just end up worse off than they started.

No, you can't blame your wife for that method.

It was my method first.

She must have stolen it from me. ;)

I get scorned for doing it this way all the time!

Heap sort vs insertion sort.