Thanks @tomgey, I appreciate you being so thorough! 
I don’t know which thread from a few months ago you were referring to, but see my point 3 in my original post for what I tried. I definitely hear the click, quote:
[I tried removing] the SIM card and re-inserting it. I can confirm that I inserted my SIM card as shown in the official video, it matches this photo by @AnotherElk , and I can hear the audible click of the SIM card snapping into place (so it is not this problem).
Give my previous post and the above, could it be a problem with my SIM card as @teezeh suggested? Or more likely a problem with the Fairphone 3 (I hope not!)?