Ok: More with this situation, as it happens.
Without any modification, I can now send emails through the smtp using Outlook. It just decided to do so.
I set up two Accounts in Outlook: one IMAP and one POP.
When I use the POP account, mail goes through the smtp, as described.
When I use the IMAP account, mail just sits there in the outgoing mail folder.
SMTP config for both accounts are EXACTLY the same, except of course for the ports: 993 for IMAP and 995 for POP. The only difference, and not relevant for the smtp (or at least I think so).
Thunderbird is still solid and sends messages with both configs (IMAP and POP, which share a single SMTP config).
Outlook is not my friend anymore, because it died suddenly. Please send flowers to the deceased software.
I still need to be able to send reliably mails through it though: our clients are 95% Outlook lovers, and they don't care so much if Outlook is dead or not
.
François