Third Party Library Plugin won't load in packaged build

Plugins created using the Third Party Library template won't load in a packaged build or the Launch button in the Editor Toolbar, but will if launched through the Project Launcher Window via the Editor.

Found in 4.22 CL#7053642

Reproduced in 4.21 CL#4753647 ,4.24 Main CL#7206954

1. Create a new project.
2. Create a new plugin using the Third Party Plugin template.
3. Click Launch button in the Editor toolbar or Edit->Package Windows->64 Bit.

Results: A popup titled "Message" that says "Failed to load example third party library" will appear for every plugin created this way.

Expected: The plugin properly loads in a packaged project or via the Launch button in the Editor Toolbar.