Ok, so now everything’s back in order! I can use my phone as a phone again! 
Here’s what I’ve done (not sure exactly what did the trick):
- took out the microphone module (the one at the bottom) ;
- cleaned all visible contacts with alcohol ;
- placed 3 layers of tape (I’d say about 1 millimeter in total) underneath the module before screwing it back in place, like explained in this iFixit comment ;
- put it all back together and called someone to check if everything’s back to normal and asking them to call me to make sure it works both ways!
If I understand the situation correctly, the shell gets a bit loose over time and the module isn’t properly connected (people have noticed the microphone worked when pinching/squeezing the bottom part of the phone).
Thank you all for your help!