This case is so weird and I have never seen such trouble during my working with the Mac operating system. Since the macOS Monterey installation successes, it’s impossible that your Mac still stays and acts on macOS Big Sur.
The reason for the unsuccessful macOS update is difficult to tell, I speculate it might be an operating system error, a poor internet connection, the hard drive glitch, or so.
Facing this issue, my opinion is that you can try to clean your Mac to remove useless caches, junk, errors, glitches, or other residue files that might result in a failing macOS update. Just download iBoysoft DiskGeeker for Mac to proceed with a clean Mac hard drive clean, which automatically collects all files and folders on your Mac in a visual table, allowing you to remove the potential and malicious file or folders being blamed for this failing OS upgrade.
The Clean Junk feature of iBoysoft DiskGeeker is designed to scan the system drive usually labeled as Macintosh HD, Macintosh HD - Data, macOS, or macOS - Data for redundant data. It can find information and files no longer needed in the system cache, user logs, application cache, downloads, etc.
Files and folders will be listed with file names, modified dates, and sizes for users to select which files should be cleaned to free up the space of the system drive, improve Mac performance, and reduce the system crash risk for a smooth macOS update.
If this way fails to help you deal with the macOS Monterey issue, try to boot into macOS Safe Mode to check if everything goes well. macOS Safe Mode is a diagnostic feature that isolates every suspicious factor making your Mac work abnormally and keeps the functions or programs essential for Mac startup. You can even update your Mac to macOS Monterey in macOS Safe Mode!
Here is how to boot your Mac computer into macOS Safe Mode based on your Mac model:
For users running a Apple silicon Mac:
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Shut down your Mac.
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Turn on your Mac and continue to press and hold the power button until you see the startup options window.
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Select your startup disk, then press and hold the Shift key while clicking “Continue in Safe Mode”.
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Log in to your Mac. You might be asked to log in again.
For users running an Intel processor Mac:
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Turn on or restart your Mac, then immediately press and hold the Shift key as your Mac starts up.
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Release the key when you see the login window, then log in to your Mac.
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You might be asked to log in again. On either the first or second login window, you should see ”Safe Boot” in the upper-right corner of the window.