Sharing version catalogs across multi-project builds is straightforward. Sharing across multiple single-project builds is not as simple. Based on what I’ve read, TOML files are not a good solution but the programmatic API is and a settings plugin might be. How would I use either of those approaches to solve this problem? Left to my own devices, I will likely create a Plugin, apply it in the settings script to add a version catalog populated with suitable data for all applications that might need it. Never having created a Gradle plugin, I could be way off base. Advice or an example to look at would be greatly appreciated.
You’ve read the wrong things then. 
You can of course do it as settings plugin, and if you do it as precompiled script plugin, it is as easy as writing a settings script itself.
But you can as easily publish a TOML in an artifact and then depend on that artifact in the other builds as documented on Sharing dependency versions between projects.