Re: The facetime app is not working correctly on MacOS Monterey 12.7.6
I guess you would have checked that your wi-fi connection is not-a-problem, and you have internet access.:
Here's Apple Support's online article for MacOS Monterey: Wondering if any of the tips in there can help:
Make calls in FaceTime on Mac - Apple Support
and / or
If you may have any restrictions on using FaceTime set: See: Turn off Screen Time on your Mac - Apple Support
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Re: "It only allows me to create "links" for Facetime"
If you haven't already:
(1) Quit the FaceTime app.
(2) Restart your Mac ie: Apple menu > Restart. (Deletes temporary files, re=establishes connections, may update any recent setting changes, and more ; often restores order if something is not working as expected) Wait for Mac to restart and log back-in.
(3) Reopen FaceTime, and see whether you can now make and receive calls.
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Re: "The facetime option is "greyed out" and I don't even get facetime calls on my Macbook pro 2016 anymore when someone tries to facetime me".
If the problem-situation continues:
If you have any unnecessary third-party (non-Apple) apps like virus-checkers, ad-blockers, cleaners, and the like (even VPN ... unless provided by an employer) installed on your Mac, try turning them off. (Better yet, uninstall them using uninstallers from their developers to clear up any associated files as well as the apps.) Such apps often cause interference.
Any third-party apps you find necessary, need to be kept up-to-date so they don't cause compatibility issues.
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Still need help?
Is your Mac short on memory / storage space ... eg: A Mac generally needs at least 50 Gb free in order to do alll you ask it to do, and to run smoothly. Check Apple menu > System Preferences (/Settings) > General > Storage:
If space is short: Remember it is wise to first back up your Mac, before making space available
Try FaceTiming someone directly from their details in their contact card ? Ask Siri to FaceTime someone ?
Try checking through Apple Support's trouble-shootiing article (link below) ?
If FaceTime isn't working on your Mac - Apple Support
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If you wish: Send feedback to Apple using their webform: Feedback - FaceTime - Apple