The "non working" systems were missing the following files in the "bin" directory:
"App_global.asax.compiled"
"App_global.asax.dll"
We believe it was due to virus scans happening on the on-premise Devops server during the publish step of the pipeline. This can prevent the precompiled files from being generated, without any warning or errors.
http://ydie22.blogspot.com/2008/11/aspnetcompiler-and-missing-compiled.html
The solution we chose was to stop precompiling during publish. The website was using an output type of "Class Library" and was being used as a Web Application, so we believe the precompile settings were a leftover from before it was converted to a Web Application, and has been limping along (but working) for years, until now.