Why doesn't Time Machine do scheduled backups after my Yosemite -> El Capitan upgrade? Summary

Since upgrading to El Capitan from Yosemite, Time Machine works manually, but automated backups and disk icons are no longer working as they should.

Detailed Problem Description

I have a brand-new system that came with Yosemite, and Time Machine was working fine. After a recent upgrade to El Capitan, I have the following issues:


The Time Machine disk (an external USB 3.0 device) no longer shows up on the desktop as a Time Machine device. It shows as a standard removable disk.


Even though Time Machine sees the device and schedules periodic backups, these backups never run. The only backups that run are the first one scheduled when the Time Machine disk is mounted, or when "Back Up Now" is selected. In this example, the manual backup I ran at 4:00pm completed, but the incremental scheduled for 4:59pm never ran.




Troubleshooting Steps

On my own, and with Apple technical support, I did all of the following in an attempt to debug:


Sensible things I tried on my own:


Perform First Aid on the Time Machine GUID table.
Perform First Aid on the Mac OS Extended (Journaled, Encrypted) filesystem on the disk. 
Unmount and remount the Time Machine disk.
Deselect and reselect the Time Machine disk.
Disable Time Machine, reboot, and reselect the Time Machine Disk.

Things I did at the request of tech support:


PRAM reset.
SMC reset.
Boot in Safe Mode.
Configure Time Machine as a different Administrator account.
Boot to recovery partition and non-destructively reinstall OS X.



Other than performing a full erase and reinstall, I'm at a loss as to how to troubleshoot this issue further. This seems to be a bug in El Capitan, but of course there's always the possibility that some kernel extension or third-party extension like FUSE, VirtualBox, or VMWare Fusion is wrecking havoc, but I require those extensions for work.

Apple currently denies that there are any outstanding Time Machine bugs in El Capitan, but that may be due to its newness or the fact that not enough people have reported related issues yet. On the other hand, if it's really just my problem, what else can I do to debug this or work around this (e.g. a cronjob of some kind, perhaps?) short of hosing my entire system and starting from scratch?
I'm having a very similar problem after upgrading to El Capitan, except that I can not repair the disk, my attempt always gives me a repair failed error.  I don't have an answer, but I think I can verify that the issue probably isn't related to any third-party extensions as I'm operating without.  So at least I may be able to help rule out those as a source of the problem.