While writing this question I understood what was the problem, I took care of it, and then the email was sent successfully.
To understand my answer below, one has to ensure understanding of how email data gets transferred from a CMS (WordPress in this case), all the way to the final, destined email account. This chain describe it:
CMS and its contact form (WordPress with CF7) → php's
mail() function →
sSMTP → SMTP (web
service) → email-proxy station (Gmail) → final email destination
The problem and its solution
sSMTP conf above was actually just fine. The problem came from my email proxy to which I've sent the data via
You see, after you create an email account which will serve as a proxy (a medium) between the content sent via SMTP to your final destined email address, this proxy is generally useless for
sSMTP and similar tools until you enable something in that email proxy itself !
In my email proxy, Gmail, what I had to enable is called "access for less secure apps".
I got an email to Gmail saying "Review blocked sign-in attempt". I opened the email, and understood that Gmail will approve the IP address from which this data came if if I allow it. This email address is my VPS machine's IP.
I allowed getting email sent via
sSMTP which Google deemed "less secure app". I then I tried to send an email again from WordPress. This time, it went fine.
The problem was on Gmail's side --- I had to enable "access for less secure apps" there, then the email was sent and the data reached its final destination.
- That's a bit off topic but when I first started to learn on
sSMTP I asked myself "how to ensure sSMTP` will work with more than one application. Here I got the answer at the time.