When having only Apotheosis, JEI and Tropicraft (and any respective coremods) JEI simply does not operate.
Ctrl+o does not open anything, R and U do not work; and although you can edit the control/keybinds for the Overlay, there isn't any other evidence that JEI is even installed.

I wish I knew more to tell you which mods fault it is; but I spent all of Friday narrowing it down until I learned that it was only when all three of these were together, that there was an issue.

As a note; I have reported this issue to all three mod developers, if I've totally screwed it up somewhere- I'm sorry.

Steps to Reproduce

1. Install JEI
2. Install Apotheosis
3. Install Tropicraft

Apotheosis API Version: 4.6.1
JEI Version: 1.16.5-7.7.0.107
Tropicraft Version: 9.0.0-beta+54
Forge Version: 36.1.32

No applicable crash logs.

May not be any crash logs but there should still be relevant debug.log and latest.log files containing useful information

I ran /start debug and /stop debug while opening inventory and attempting normal JEI commands.
profile-results-2021-07-03_04.21.31.txt
Though I don't think that's the debug you were looking for. I did not see a debug.log. Here's the latest.log:
latest.log

Appears that perhaps Tropicraft was the root cause? They indicated on my post there that they are making a fix.
Tropicraft/Tropicraft#395

Thank you for your quick response and consideration. Your mod is excellent, and I wish you all the best.