TL;DR It looks like your intentions were good on this one, but your wording (on the first few readthroughs) makes your comment look more like an opinion than a back-up suggestion.
Let's clear up a quick misconception here first though. You say
[W]e are urged to comment when we downvote an answer
But that's actually not true. You're urged to leave a productive, actionable comment on an answer you've downvoted if you think it's salvageable. However, if you don't think the answer can be fixed, your only options are to downvote and possibly leave your own answer explaining your opinion and giving an alternate solution. For more on this, take a look at this well written old thread.
You also say
I think it's in line with the back up policy that has been enforced more widely recently.
Again, if you think an answer isn't up to snuff per site policy, you're encouraged to downvote and flag. If you do leave a comment, it should have specific advice for how the user can fix their answer.
I've reread your comment a few times now and it seems like you may want them to remove the last paragraph (or reword it). But even then I'm not completely sure. And I think the reason for that is because you haven't worded the comment as a suggestion, you've worded it as a disagreement.
Let's take another look at the comment you left (that was deleted):
I'm downvoting this answer mainly because of the last paragraph : it seems your understanding of this disease is pretty basic (psychology 101), too basic to provide for an issue this complex. I do agree with your advice to talk to a professional to get useful tips.
To reviewers, the OP, and even to me, this reads as "I disagree with your answer." which is not too productive. However, if instead, the comment had been
I agree with your advice to talk to a professional to get useful tips, but given your experience with the subject (as stated), I think your last paragraph may do as much harm as good. I'd suggest removing it or providing more experienced sources to back the information up.
I can almost guarantee you it wouldn't have been deleted.
Especially since we're a site where people answer and comment with their worldviews and personal experiences, it's very very easy to start arguments. This is part of the reason for our strict comment rules. We've had a looooong history of pretty ugly lengthy debates in the comments and it's not something we'd like to repeat.
When you're commenting in the future, make sure you leave a clear actionable suggestion if you have one and your comment will be safe (at least until that suggestion is acted on, in which case the comment will be No Longer Needed and likely removed).