The riddle has been solved, at least for my company. The short answer to why I could not open a Notebook in ArcGIS Pro is because of our proxy. More specifically, we have been adding User Variables of http_proxy and https_proxy to our computers’ Environmental Variables.
This was done in order for us to be able to access the AGOL via the ArcGIS Python API. It works just fine in Pro 2.9.x…but the presence of the http_proxy and https_proxy variables specifically prevents Pro 3.x from opening a Notebook. I did multiple trials and either removing these variables or having these variables with a single space allowed a Notebook to open.
Unfortunately, the ArcGIS Python API still needs to have the proxy set in order for us to connect to our AGOL, but we are able to pass that info through to the GIS() object as a variable. But, at least the mystery has been solved and we are able to use Notebooks in Pro 3.x.
TL;DR - Adding proxy settings to my Environmental Variables caused Pro 3.x to throw the "Failed to load notebook" error when trying to open a Notebook. Deleting those variables fixed the problem.