UPDATE:I am running OS X 10.11.2
Beta and with the second update of that version Time Machine began to once again run automatically.
The El Capitan 10.11.2 Public Beta improves the Stability, compatibility, and security of your Mac.
The focus areas for this update include
- Graphics
- Mail
- WiFi
- Calendar
- USB
- Notes
- Photos
- Spotlight
Apparently somewhere in there a fix for the Time Machine problem was included.
If the Time Machine drive is connected by USB, USB may be the issue. I have an Apple USB keyboard and a USB Apple mouse plugged into the keyboard. After upgrading from Yosemite to El Capitan, the keyboard and mouse would sometimes refuse to power up. This happened when trying to wake the computer from sleep by either pressing a key or clicking the mouse. The only recourse was to force a shutdown and then restart.
I did a lot of troubleshooting, including trying different USB ports on the computer, a different keyboard and mouse combo, and resetting SMC. My wife finally nailed the problem (and she's inordinately proud of this since she's not computer-savvy) by pointing out that I had a powered, 4-port USB Belkin hub plugged into a USB port on the computer. No, I didn't plug the keyboard into it (although I tried that once when the keyboard wouldn't respond) but after disconnecting the hub, the problem never recurred.
My guess is that the USB issue listed by Apple in the 10.11.2 beta will address this. After the final update is available, I'll try the hub again to see what happens.
My Time Machine backup is not affected because it is connected via Thunderbolt, so I have had no issues with it.