"This copy of the Install macOS Monterey.app application is damaged, and can‘t be used to install macOS."

I've had similar problems and have had better luck installing from the Safe Mode because I apparently have some 3rd party software that interferes with the installation so try the following:

#1 - boot into Safe Mode according to Start up your Mac in safe mode - Apple Support.

NOTE: Safe Mode boot can take up to 3-5 minutes as it's doing the following; 

• Verifies your startup disk and attempts to repair directory issues, if needed

• Loads only required kernel extensions (prevents 3rd party kernel/extensions from loading)

• Prevents Startup Items and Login Items from opening automatically

• Disables user-installed fonts 

• Deletes font caches, kernel cache, and other system cache files

#2 - go to the System/Software Update preference pane and begin the download.

#3 - run the installer from the Safe Mode.

I ran into this the other day and it was very frustrating. Here's how I worked around the problem:

Reboot the Mac (might be residual mounted disk images from earlier attempts)

Download Monterey so you have /Applications/Install macOS Monterey

Open Terminal:

sudo /Applications/Install\ macOS\ Monterey/Contents/Resources/startosinstall --agreetolicense

Enter your password and Terminate the Terminal when prompted. The install should actually work.

I have no idea what checks or balances are failing but it worked fine when I ran it in this automated command line method.

Yours is not an uncommon problem. What has typically happened is that the file got corrupted during the download and that Wi-Fi was used. Wi-Fi will often fail for these very large files. It is too slow. Instead use a wired Ethernet cable connected to your modem even of you have to move the computer temporarily. Then do a clean new download and it should do just fine. Good luck.

You merely open the Terminal App and copy and paste that line that starts with sudo /Applications/... and then paste it in the terminal. Then enter your password. The command runs a program inside the Monterey App to run the installer and apparently, it works around the error.