From those points it suggests that the isue is outside your browser (as it affects all browsers regardless of browser profile, cache, etc.).
You have not listed any external services yiu are using, so I assume therefore you have none.
You do not appear to have tried the fourth test, so I cannot tell if there is an issue in the OS, or that is related to your specific user account. Hopefully, you will still consider testing that.
It is a standard app in Monterey.
When you are following an
https://www.icloud.com/shortcuts/link, you are being taken to a web page on your Mac. When I follow such a link, it is automatically being passed to Shortcuts for download. I don’t know if this has to use Safari (or another browser) to initiate this, or if the OS is the key link. But, the initial behaviour is what is in error here. You should never get to the web page with the download link, you should just be taken straight into Shortcuts.
My best guess would have been some sort of plugin, VPN or security protection software intercepting it, but you don’t seem to have anything you think could affect the URLs you are following.
Out of interest, is there any change in behaviour if you try running this terminal command to open the URL rather than following the link from within the browser?