Why am I unable to install macOS 12.5.1 on my desktop computer? I have an iMac (21.5-inch), purchased in late 2009.

A late 2009 iMac cannot be upgraded beyond 10.13.6. See the following from Identify your iMac model - Apple Support

iMac (21.5-inch, Late 2009)

Model Identifier: iMac10,1

Part Number: MB950xx/A, MC413xx/A

Latest compatible operating system: macOS High Sierra 10.13.6

In short the hardware is too old and does not have the components to run any OS beyond 10.13.6. Have you considered replacing the computer, after about 14 years it's time!