I know this is an old thread but, hey, it's clearly something Apple still haven't worked out or got on top of in all these years.
The key point about this issue is that FaceTime is documented, in Apple's own documentation, stating that a camera is only required when doing FaceTime video calling. The prerequisites for FaceTime are only a working microphone and speakers, allowing you to do FaceTime audio calling as well as Continuity cellular calling, with your iPhone.
This seems to be one of those bugs/inconsistencies, wrapped up in a perception filter at Apple, that keeps on causing problems for people, especially those of us who use Macs with no camera (Mac Mini and Mac Pro or Macbook Pro in clamshell mode) who try to use Continuity cellular calling, despite Apple's best efforts to mess with it. There is a plethora of posts about this issue across many fora, across many years. I have posted on this many, many times, over the years and even had a support case escalated within Apple where the outcome was, "This is by design". If your intent is to design badly then I would wholeheartedly agree, with that.
Interestingly I have had it work for extended periods but it seems to break with each new major release of macOS. The most recent break being when I upgraded to Monterey, having used FaceTime, with my Macbook Pro in clamshell mode, for months but only after many months of it complaining about no camera, after upgrading to Big Sur.
Whatever Apple have done "by design" seems to keep on coming back to haunt them, after they "fix" it!
More here: https://forums.macrumors.com/threads/use-facetime-when-mbp-is-in-clamshell-mode-2020-mbp-fyi.2240141/
