Well, I guess if I can't get an answer, I have to go elsewhere. I would have liked if someone had clued me that Outgoing SMTP Server area will accept the text equivalent. I assumed it would, I now know it will.
This is what turned out to be the issue: The client is migrating to Google Mail (gmail). FileMaker accesses the SMTP server directly so you have to put "smtp.gmail.com" in this area (or whatever your new SMTP Server wants). In addition, gmail requires different ports and SSL. Google publishes a doc on this. Good luck!
You may have just helped out others. I've seen scattered reports here that suggest that gmail is working for some, and not for others. The information you've just provided may explain why this has been so.