If you join a multiplayer server directly after minecraft has loaded, the JEI GUI does not appear.
If you start minecraft fresh, join a singleplayer world first and then a multiplayer server the GUI appears and works normal.

Pretty sure the cause is IE.
I just put the newest IE version in my mods folder and the problem appeared right at that moment.

Forge Version: 1.16.3-forge-34.1.42
IE Version: ImmersiveEngineering-1.16.3-4.1.1-127.jar
JEI Version: jei-1.16.3-7.6.0.49.jar
JEI Integration: jeiintegration_1.16.3-6.0.0.8.jar

Full modlist:
modlist.txt

So in the 1.16.4 Version of IE it seems like this Glitch/Bug is not existent anymore.
Think this issue is closed cause of this. :)

Please provide us with your debug log file

Hy I attached both debug logs.
One where I joint mulitplayer directly without JEI loading and one where I first join SP to get JEI working and than MP.
I'am sry I didn't know exactly what you need to debug it. If you need more please let me know. :)

THX for your effort!

debug_direct_join_to_multiplayer.log
debug_first_SP_than_MP.log