My MacPro 2013 won't update to Monterey when I HAVE the genuineApple media

Here are the minimum computer requirements for Monterey:

macOS Monterey is compatible with these computers - Apple Support

Is your Mac Pro listed as Model Identifier: MacPro6,1? It does show Monterey is the latest OS for that model number.

I have seen that reported on a non-Apple SSD, but nothing in the Apple Support community for a solution on a genuine Apple SSD. Here is one from Reddit, if that helps at all.

https://www.reddit.com/r/MacOS/comments/s25a17/a_required_firmware_update_could_not_be_installed/

You could try making a bootable USB thumb drive from the install application. Instructions here:

Create a bootable installer for macOS - Apple Support

name your BOOTable thumb drive exactly MyVolume

use the Installer sitting in the /Applications folder

copy the ENTIRE command-line for Mojave from the article (it extends beyond the edge of the page) and PASTE at the Terminal prompt and press return.

Although it is not an Apple official recommendation, some users have suggested that installing MacOS 10.14 and fully updating it will provide a more advanced firmware (version 131.0.0.0.0 or higher) that stops being fixated on the Apple Brand SSD.

That is worth a try.

The version of firmware you are showing was installed with High Sierra updates.

No need to go backward. Install that Mojave on ANY Drive, even an external USB drive.

The firmware is saved in private store on the processor module, not on any drive. Firmware persists across drive failure.

Do you still have the original Apple SSD which shipped with your Mac? If so, temporarily swap SSDs internally so you can install macOS Monterey to the original factory SSD. Then once Monterey is installed....the system firmware will be updated to the Monterey version. Swap SSDs back again & try upgrading your Big Sur installation on the 1TB SSD to Monterey.

FYI, if that 1TB SSD is not meant for this Mac Pro, then it is possible macOS knows this & is refusing to install the system firmware update because macOS does not see this 1TB SSD as valid for this system. Yes.....macOS may not work correctly if the SSD is not correct for the computer. I had an Apple OEM SSD installed into a different model Mac and macOS would not boot the computer while it was installed even though the SSD worked fine when booting & using Linux.

JimB-Dallas wrote:

My Adobe Lightroom and Photoshop will not update because they require Monterey.

Surprisingly enough, you can still install the current version of Photoshop on Monterey. But there is a footnote that you may have missed.

Adobe – Photoshop system requirements

Adobe – Camera Raw system requirements

"Adobe Camera Raw (version 17.0 and later) is not supported on macOS 12.x; we recommend updating to a later version of the OS for ACR support."

Adobe – Lightroom system requirements

Adobe – Lightroom Classis system requirements

Both Lightroom applications now require Ventura (13.1) or later.