The fact that both problems appeared after updating to macOS 27.0.1 is certainly worth noting, but it does not by itself show that the update caused them.
Apple’s published information for 27.0.1 does not identify a built-in-camera or automatic-restart problem. Apple’s macOS 27 update information
For the camera, start with System Settings >> Privacy & Security >> Camera and make sure the apps you are testing have permission to use it.
If you use Screen Time, also check its camera restrictions. Apple includes both checks in its camera troubleshooting guidance. Apple’s camera troubleshooting guide
Then try the built-in camera in Photo Booth and another camera-using app. If you get the same black image in both, the problem is less likely to be specific to one application.
The next useful test is Safe Mode.
On your Apple-silicon MacBook Air, shut it down, hold the power button until the startup options appear, select your startup disk, hold Shift, and choose Continue in Safe Mode.
Apple says Safe Mode can help determine whether a problem is related to software that loads during startup.
Test the camera again.
If it works in Safe Mode but not after a normal startup, that gives you a useful direction for further troubleshooting.
If it remains black, Safe Mode does not by itself identify the cause.
If it still fails, testing the camera from a new macOS user account is another useful check.
If it works there, the problem may be confined to your existing account; if it fails there too, that makes an account-specific problem less likely.
For the other problem, disconnect your external devices, hubs, drives, displays, printers, and other USB or Thunderbolt devices. and see whether the behaviour continues.
Apple recommends removing peripherals when investigating unexpected restarts. Apple’s unexpected-restart guide
It is also important to establish exactly what the Mac is doing. I
f you choose Apple menu >> Shut Down, the Mac completely powers off, and then starts again by itself, that is an automatic power-on after shutdown rather than simply an unexpected restart.
If instead it restarts while you are using it, check Console for a recent panic or shutdown-related report, if one is present.
Also check System Settings >> General >> Software Update for a newer macOS release.
If one is available, installing it may be appropriate, although that does not mean it will necessarily resolve either problem.
Since you have already reinstalled macOS and run Disk Utility,
I would not repeat either one for now.
If the camera continues to fail in multiple apps, Safe Mode, and another user account, or the Mac continues to start again with peripherals disconnected, those results, along with your Apple Diagnostics information,would be useful when contacting Apple Support or an Apple Authorized Service Provider. Apple recommends service when the built-in camera remains unavailable after its troubleshooting steps.
Have the computer evaluated by the Professionals who have the Special Hardware Software and Expertise to diagnose the issue and offer possible solutions
Make an Appointment at the Apple Genius Bar or call 1 800 MyApple and have them evaluate the computer
Other location Call Centres
Contact Apple for support and service