Thanks for your efforts! I emailed Google's Nexus support and was pleasantly surprised to get a response.  Here is part of what they said:

If your device is having camera issues, it could be an issue with a 3rd party app or something else that was installed. Here are three steps to stop the camera app from working improperly:

Step 1: Check for system updates and app updates.
- To update the OS, go to 'Settings' > scroll down to 'System' > click 'About phone' (or tablet) > click 'System updates'.
- To update apps, visit the Google Play Store app.

Step 2: Try using the device in safe mode (and uninstalling apps in safe mode). [ http://www.gottabemobile.com/2012/08/02/how-to-boot-the-google-nexus-7-tablet-in-safe-mode/ ]
- Hold the Power button while the device is already on to get the "Power off" option.
- Press and hold "Power off".
- Select "OK" to reboot into safe mode.
- To uninstall an app, go into "Settings" and then under the heading "Device" select "Apps".
- Select your installed apps and there will be an option to uninstall.

- To exit safe mode, power off the device normally and start it back up normally.

Step 3: Perform a Factory Data Reset

**NOTE that all of your personal information will be removed from the device. Please backup all of your information first.**
[ Here is one suggestion how http://www.dummies.com/how-to/content/how-to-back-up-your-nexus-7-information.html ]
- Go to Settings > Backup & reset.
- Select Factory data reset.
- Select Reset tablet.
- If your device is password protected, you'll be prompted to enter your Pattern, PIN, or Password.
- Device Erase everything to confirm.

I considered removing the camera app but I paid for it.  I'm a newbie, so I was afraid that I'd have to pay for it again upon reinstall.  I'm not sure how the Google Play store works.

(I just educated myself - found out that I don't have to pay for it again!  Good news
https://support.google.com/googleplay/answer/113410?hl=en-CA )

Will try the uninstall tomorrow and report back

Interesting.  I'm on 4.3 and have tried both "Nexus 7 Video Camera" and "Full HD Video Recording" apps and neither has worked.  They don't even successfully launch.

(I got the impression from somewhere that the video interface changed for the 2013 version and many apps don't work).

Could you give me the names of your working apps?

Thx