After a few tries I did get this method to work. But I'm going to post exactly what I did to hopefully help neophytes like me who need all the steps.
In ArcGIS Pro, I went to the Analysis tab and clicked "Python" in the Geoprocessing section.
In the New Notebook that opened, I entered the following bold text, specific names are in brackets and should be replaced and brackets removed.
import arcpy
in_project = "[project name].aprx"
output_file = "[output file path, with forward slashes, e.g. C:/ArcGIS Projects/etc].ppkx"
arcpy.management.PackageProject(in_project, output_file)
Then I hit "run" and it gave a very helpful set of error messages after it finished.
Like others, it was custom tools (which I told it to exclude) that were the problem.