I have found the solution, with a little help from the Siteground technical support. It is not all that obvious at first, but makes sense in the end.
One must set up at least one bonafide email account on the site, so as to be able to authenticate with Siteground's SMTP. (The native php method does not work, as cron.php does not have a sufficient level of permission to run it).
Assume a domain abc.com
an email account firstname.lastname@example.org
with password yyy
Here are the Email settings that matter:
SMTP hosts: abc.com
SMTP Username: email@example.com
SMTP password: yyy
(Now here's the tricky part!)
No-replay address: firstname.lastname@example.org ( I am not certain it must be the same, but it must be a bonafide email address on your domain.)
Support name: email@example.com ( I am not certain this field figures in the outcome)
Support email: firstname.lastname@example.org
Why these two other fields? Apparently the info from these fields (Default being: email@example.com and firstname.lastname@example.org) finds its way into the FROM: field of the mailer. But Siteground does not allow its SMTP to send out mail whose FROM: address is not a bonafide Siteground email address (I am not 100% certain it has to be from the same domain, but I would guess so.). Such a policy probably prevents people from using their SMTP to spoof other people's email.
This policy may have been somewhat recent. I seem to recall my Moodle on Siteground could and did sent out forum mail in the past. I did a workshop for beginners and came home to find tons of email messages from their forums. My support email address was one not on siteground. That is probably what killed it.
I hope this will help others.