This happened before, so I thought it was a senior moment.
Where did my wallpapers go?
This is the wallpaper that I added to the "other guys."
But as you can see, it ain't there no more.
I have a copy in a directory called /home/andy/My_Wallpapers/.
My unclear first post was due to the dam* cognitive impairment. 
click on Add (bottom right, green), navigate to your wallpapers, and select it.
alternatively: right-click on the picture and select "set as wallpaper"
I edited my first post.
Since I have not gotten a response that indicates an understanding of what I would like happen, I will do a "MacGyver."
Can I give my wallpaper an immutable attribute so it does not disappear?
Pavlos gave you the "proper" answer, regarding setting up such that, when you click on "Add", it would remember the last place you selected your wallpaper from.
So, on the following, click where it is circled.
Then navigate to where you have your wallpapers, i.e.
Whichever one you choose, will become your wallpaper.
Why you wallpaper(s) disappeared from the settings could be a result of your having reinstalled (not upgraded) your OS, and MATE reset itself to defaults.
Otherwise, if you either renamed your directory from something else to My_Wallpapers, or you renamed the wallpaper file itself, the Preferences would determine the "wallpaper resource" (i.e. your previously selected wallpaper) has an invalid PATH definition (MATE working with pointers/references, not with actual files), so it automatically removes the offending reference, hence it going "poof" from the choices presented.
Also if you move the wallpaper to a different folder it will disappear. Say you just have it in the Pictures folder, then decide to organize and make a Wallpaper folder and move it that folder, it will disappear until you manually locate it again. I learned that through experience. Again the path is then wrong.
