Can’t Install Monterey or Ventura on iMac WITH Apple SSD

The first problem is that you are out of disk space. Your screenshot shows only 33.93GB left out of 1TB (3% free space). That's hardlly enough space for macOS to even run, and not enough space to install much of anything. My guess is the "required firmware update" failed because there isn't enough available disk space. I know 33 GB sounds like a lot of space, but it's not, especially when you attempt to upgrade macOS. You really should not let your system drive get above about 80% full; take 150-200GB off the drive and try the update again.

If you are attempting to run the installer while booted into macOS, the perhaps you have some third party software installed which is interfering with the normal operation of macOS. Try booting into Safe Mode and trying the upgrade.

Or create & use a bootable macOS USB installer.

Create a bootable installer for macOS - Apple Support

Try running Disk Utility First Aid on the physical SSD. If that runs clean, then run First Aid on the hidden Container. Within Disk Utility you may need to click "View" and select "Show All Devices" before the physical SSD and hidden Container appear on the left pane of Disk Utility. Even if First Aid says everything is "Ok", click "Show Details" and scroll through the report to see if any unfixed errors are listed. If so, then run First Aid again until the errors are gone (may take several attempts), otherwise you will need to perform a clean install of macOS by first erasing the whole physical SSD (Intel Macs only) followed by reinstalling macOS & restoring from a backup.

It is possible the hard drive is failing. Run DriveDx (free trial period) to check the health of the drive and post the complete DriveDx text report here using the "Additional Text" icon which looks like a piece of paper on the forum editing toolbar.

FYI, according to MacTracker, the last supported version of macOS for this iMac is macOS 13.x Ventura.

Make an appointment at the Genius Bar of your local Apple Store.