Just to add my experience in my recent international travel with an expired green card+ extension letter in case this helps anyone.
I flew from Dallas, TX to Algeria and back transiting through CDG (Paris, France).
Air France staff in Algeria was not familiar with the extension letter, it took them some time to read it and confirm with 2 different people to finally get it. I went through security and border police who also were not familiar with it. The police officer took them and went to confirm with a someone else, they took pictures of the GC and extension letter then let me go through.
In CDG (France), air France staff checked it again but were much quicker and familiar with it, no questions asked. Arriving at DFW, it went smooth as well, the officer took a look at my GR and extension letter and that was it !