Your M2 MacBook Air is compatible with macOS 27, so the age of the M2 processor does not by itself explain the problem.
Apple lists Apple Silicon Macs as supported for macOS 27. (Apple Support)
Because the menu bar stopped responding after the macOS 27.0 update, the timing is relevant, but it does not establish the cause.
First restart the Mac and check whether the menu bar responds.
Also check whether the problem affects all menu-bar items or only particular items; this helps distinguish a broader macOS interface problem from an issue involving an individual menu-bar item or application.
If the problem continues, temporarily quit or disable third-party applications that add menu-bar items or other system-interface modifications, then test again.
If necessary, check System Settings >> General >> Login Items & Extensions for software that starts automatically.
If the problem remains, start the Mac in Safe Mode.
Shut it down( Do Not use Restart ), press and hold the power button until the startup options appear, select your startup disk, hold Shift, and choose Continue in Safe Mode.
Apple states that Safe Mode can help determine whether software that loads during startup is contributing to a problem. Apple — Start up your Mac in safe mode
If the menu bar works in Safe Mode but not after a normal startup, software loaded during normal startup becomes an area to investigate.
If it remains unresponsive in Safe Mode, the problem is less likely to be caused by those startup items, and the next step would be to investigate macOS itself rather than assuming the M2 hardware is at fault.
An aside. I have a M2 Mac Mini running macOS 27.0 Golden Gate. The machine itself was not too old for the upGrade and No I do not experience the issue you have outlined