Just tried another SMTP server in the plugin. The "test" email makes it through just fine, but there is no automatic e-mail as should happen in my contact form. I don't see any error messages at the SMTP side or any bounce back messages. It's like contact form didn't even try to send it.
Then I tried to change to a new SMTP plugin (https://wordpress.org/plugins/configure-smtp/), and now I am getting results.
Result: There appears to be a conflict with the original SMTP plugin
I should also note the following warning shows up on the screen if the message is initiated by a user not logged into WP:
Warning: array_keys() expects parameter 1 to be array, string given in /home4/ab3585/public_html/wp-content/plugins/jetpack/modules/contact-form/grunion-contact-form.php on line 1476