I looked harder at your solution and have the following to report.
Using Sheet Set Manager, there is an 'Import Options...' button in the Choose Layouts step. One of the options is to, "Prefix sheet titles with file name" so, naturally, I unselected this since that is exactly what I am trying to exclude.
I selected my files and options (anyone reading this should be aware that you have to set your publish options again here. The options you set using Publish do carry over).
I saved and went to the Publish button. From here, there are several options (DWF, PDF, etc.) but I did not see an option for using the named Page setup that I have defined in each drawing. I like to define and name a Page setup for each layout so I know it will print correctly and to scale. Anyway, I selected Publish to PDF and it plotted correctly except that it put a number in front of each file name (the number was the sheet order within the sheet set). So, I still would have had to rename the files.
I am new to using Sheet Sets so maybe there is something else I could have done but I found a better solution using the publish command.
When using the publish command:
1st try - I did not realize I could change the name of the output here. The field is titled, "Sheet Name" and is telling the publisher which layout to print. I would have thought if this was a name that did not exist in the drawing, it would produce an error but I was completely wrong. It does output a PDF with the layout name only (or any other name I specify). However, this is not much different than changing the name of the file afterward but, still, a step in the right direction.
2nd try (and my preferred solution) - Execute the Publish command. Select 'Add Sheets'; now in the Select Drawings dialog box, un-select the 'Prefix sheet title with file name...' option. From the 'Include' pick list, select 'Layout' only. Now select 'Publish' and the output files will have only the layout name; no renaming anything. Also, this sheet list (*.dsd file) can be saved and the settings will all be retained.
Thanks for making me try harder.