Thanks Nino, I appreciate the investigation and the honesty about the root cause.
To be blunt, though: from a customer's perspective it doesn't really matter which internal team owns the bug, or whether it originates on-premises or service-side, or something else. As long as Microsoft supports and allows Centralized Mail Transport in an Exchange Hybrid setup, this behavior breaks a supported configuration, and that makes it Microsoft's problem to fix, regardless of internal ownership.
For what it's worth, we raised a separate support case and were told this is a confirmed, known issue with a fix currently being deployed, plus a follow-up next week to verify it reached our tenant. So the fix does seem to be in motion. Given how many people are tracking this exact thread, it would help everyone if that status were reflected here too, even just a short "known issue, fix rolling out", rather than pointing us all to individual support tickets.
Also, why is there nothing to see in the Service Health Portal, like Krusty_the_admin is asking...?
In the meantime, the workaround we were given is to add the shared mailbox in Outlook as a separate account, so sent items are stored in the shared mailbox itself.