searching for update never finds one
Trying to update but continues "checking for updates"
iMac 21.5″, macOS 11.6
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Go to Apple menu -> About This Mac. If you are running 11.6 as your profile suggests, you should be running 11.6.3.
If you want to get Monterey, check the age and model to see if you are compatible:
https://discussions.apple.com/docs/DOC-250004301
The App Store is the only place to get Monterey itself.