I have received feedback from MS on this via a support ticket. Here's what they said and fixed:
Browse to the hidden list: https://[domain].sharepoint.com/Lists/SharePointHomeOrgLinks/AllItems.aspx
Check if default SharePoint Groups of the list are missing.
- SharePointHome OrgLinks Admins SharePoint Group Full Contro
- SharePointHome OrgLinks Editors SharePoint Group Edi
- SharePointHome OrgLinks Viewers SharePoint Group View Only
If the groups are missing, then access _layouts/15/permsetup.aspx and click on OK without making any changes.
After doing this, the Edit link reappeared. Don't know how those groups would have lost their perms; didn't even know they existed!
However, the links I added via SP Admin->User Profiles->Manage Promoted Sites, after the changes to the SP home page were instituted, still were not displayed on the SP home page. They said they're working on this and the fix was imminent. Unfortunately, I have a bit of "I'll believe it when I see it" perspective on these kinds of statements... In the meantime, the workaround is to added them via the Edit link on the page and that they'd be visible to all users, not just me. I haven't had a chance to test that statement yet and, for the moment anyway, added the other links assuming its accuracy.