We have a failure to communicate. Lets try again...
Please do.
I have installed the add-ons.
Yes, you have.
As an example I provided two screen shots.
Yes, you have.
The second screen shows the actual equipment profile with the add-ons included.
Yes, it does.
These add-ons are not shown on the right side of the add-on profiles as shown in my attached screen shot attachment #1.
True.
Obviously the add-ons got to the equipment profile somehow -- it was because I added them.
Yes. True.
I have done everything I can to demonstrate an inconsistency.
Not exactly. You've shown us your experience.

And there is a solution.
BeerSmith is actually a collection of flat files, which means that data is held in context to its use and often stored in multiple places. For instance, your ingredient files are independent of identically named ingredients in your recipes. Same with equipment, mash profiles and so forth.
Occasionally users create situations where BeerSmith overwrites files. Most often, it's trying to have two copies working on the same files or from the same folder. Other times, a random error can creep in and cause what you're seeing or something similar.
Make sure BeerSmith is closed and has fully written all files (takes about a minute on my computer) In your .../Documents/BeerSmith3 folder find a file named "plugin.bsmx" and
delete it. This file got overwritten somehow, we're going to have BeerSmith write a new one.
Start BeerSmith. You should now see all of the plug ins that you have properly listed. When you close BeerSmith, it will write a new plugin.bsmx file.
And all will be right with the world.