Thank you kenniebee.
When I was notified of your post, I was reading this thread about iMessage and FaceTime activation problems. I had just rebooted my iPhone (which was the first step of your solution) so I tried resetting the network settings again after a reboot as you suggested. FT still wouldn't sign in.
Thankfully, I got it to work a little while later. I'm not quite sure whether your solution took some time to work, or if it's the following steps I took that fixed the thing.
I went back to the thread I mentionned above and tried the solution I found. I had to adapt it a little bit because I didn't see the exact things it mentionned. Here's what I did:
- (on my PC) Logged in to Apple's "Manage your Apple ID" site (https://appleid.apple.com/)
- Clicked "Manage my AppleID"
- Clicked on the "Phone Numbers" option
- My iPhone number was listed as the "Daytime Phone". I switched it to the "Mobile Phone" field and saved changes.
- Went back to my iPhone and Facetime was signed in, ready to call
- Tested FT successfully
I'm skeptical about this. It seems too trivial to be the solution. It may just be that it didn't work right away after resetting the network settings. It tried to reconnect automatically while I was messing around with my AppleID on my PC and it was good to go when I picked my iPhone. I don't know. I don't care. It works. I'm a happy man. But just in case it may help someone else, I prefered to document what I did before it worked.
Thanks everyone for your help!