I found something that works for me and hopefully it will work for others too.
- Without your monitors plugged in, right-click on your desktop, hover over "View" and turn off the feature "Align icons to grid"
- Plug in your monitor and proceed only if your icons remain in their original positions
- Turn "Align icons to grid" back on the same way you turned it off.
Using the steps above, this should allow you to unplug/turn-off monitors and replug/turn-on monitors without your icons moving all over the place.
I noticed with this, however, that it's save-file-like and you will need to redo the steps if you want to move icons around. I tested this by unplugging my monitor, moving an icon (without changing the grid feature) and plugging my monitor back in. I observed that the icon returned to its original position on my desktop as it was before I moved it and the rest of my icons stayed in the same spot. I then unplugged my monitor and found that icon to move to where I originally moved it to, so I turned the grid feature off, plugged in my monitor and then turned the grid feature back on and found the icon to stay in its new position.
I don't know how well this will work for your games (worst off you just need to leave the grid feature off) but it will hopefully solve turning monitors on and off.