The message "fsroot tree is invalid" indicates that there is an issue with the file system root (fsroot). The file system root is the top-level directory in a file system where all other directories and files are organized.
This could be due to corruption, incorrect configuration, or other issues that prevent the proper functioning of the file system. To fix the error and install macOS Monterey, I recommend you use an external macOS Monterey bootable installer and erase your Mac. Take the following steps:
- Create a bootable installer for macOS Monterey.
- Connect it to your Mac mini.
- Ensure your Mac mini is turned off.
- Press the power button and immediately hold down the Option key until you see startup volumes.
- Select the volume that contains the bootable installer nd press Enter/Return.
- Choose your language if asked.
- Click Disk Utility > Continue.
- Click View > Show All Devices.
- Select the top-level physical disk and click Erase.
- Name the drive.
- Choose APFS as the format.
- Keep GUID Partition Map as the scheme.
- Click Erase.
- After the erase process is done, close Disk Utility.
- Select "Install macOS Monterey" and follow the instructions to install it.
Besides, your Mac mini may have hardware issues. Try running Apple Diagnostics on it.