Okay I finally understood the issue... I didn't see the side icons on the favorite menu on the screen of my smartphone 😅
If checking & fixing broken packages did not help, here is something else you can try:
For Budgie, could you try to restore the panel defaults ?
For MATE, go in Appearance -> Customize -> Icons. Try to select something else then the icons set you normally use to force the system to reload them.
Still in Appearance you can navigate to Interface and uncheck and check the Show icons in menus box
If it doesn't help after checking again your system is fully up-to-date (which is already the case according to what I understood), you can run the command sudo usysconf run -f