Thanks to all for continuing the sleuthing after my original post.

A colleague at work spend some months working on this with Microsoft and our corporate specialists. Opening documents in web/, on a tablet or phone (I understand the latter two automatically open in browser mode) has been determined definitely to cause problems. I suspect that even quick slip by someone can effect the damage to the autonumbering.

He has access to some log (SharePoint admin?) that allows him to see a lot about document history and who has done what. I don't know if it includes seeing if someone has opened in browser mode.

The separation from the template appears to be a contributing factor, for which no solution appears on the horizon. And our experience indicates that the more users are in the document, e.g., 5-7, and the larger the document (ours are often 100 pages and more and sometimes weighty with graphics), the more likely a problem is to erupt, whether that comes from the amount of activity, pasting formatted text in directly, and/or using the ribbon bullets or numbers.