OK, here's the update.
I carefully entered the SMTP settings for Gmail and Fastmail, two email providers where I have accounts. Neither of them worked, and I began to suspect it had to do with the two-factor authentication I have set up for both. For example, I created an app-specific password for Fastmail and still wasn't able to get anything through.
In desperation, I tried setting up SMTP for Comcast which is my ISP. I never use their email service, and there's no two-factor in place so a regular username/password is all I need for the ISY's notification configuration.
Bottom line: Comcast worked straightaway. That supports my original theory that the two-factor authentication was the complicating factor. I'm going to go back to work on the Gmail config and try again to see if I can get it working. I'll let everyone know what I find out.