Update:  I have finally figured it out and submitted.

Looks like the problem in this particular case was the size of the attachments.  I've had two very large Form 1099 files (so many stock sales 🙄) attached (one was over 11 MB).  Once both files were saved as Reduced PDF files and re-attached (the larger one is now <2.5 MB), the return went through. 

I'm not sure what's the max size of the attachment that is allowed in ProSeries, but it would be nice to get an error message pointing out what's wrong instead of just seeing "Return not transmitted".  Or, the system shouldn't allow to attach the files over the limit.