From testing as I developed this, I noticed that certain emojis often lead to complaints from GPT 3.5 about how the combinations were too generic. After some really basic testing I ended up removing the 3 worst offenders from the emoji picker: ['+1','-1','okay_hand'].
country flags are enabled - the search displays the Italian flag if I type in "italy". However if I type "italian", it doesn't match to the flag unfortunately.
They weren't there for me on Windows. Another one I found missing is the superhero emoji. (I'm not missing those emojis on the system, because elsewhere they work.)
Now I'm on macOS and I don't see anything missing other than the ones that you intentionaly removed.