I used the “old way”.
My sent mails still seem to be there. At least there are many many more in the Maildir/.Sent folder under my account than are shown in any mail client. And the dates of the entries go back to the early 2000s. That seems about right.
The owner of all the emails and folders is 5000:5000 which is the same as for all the other folders and mails.
That is quite reassuring. It just begs the question: Why can’t I see them in any mail client?