If Backpacked is present, wings don't render on the player, and the flight function does not work at all.

Wings work with Backpacked for me:

There must be a different mod you have which is incompatible, which is out of my control, some mods are implemented poorly and do not support mods which attach things to players. Ido is one mod which has this reported to but there is no fix currently.

I do really use a lot of mods (no Ido among them though), and I started deleting them one by one, oddly enough, flight worked back after deleting Backpacked. Switching to Wearable Backpacks solved the problem too, that's why I wrote.
Just double-checked. Deleted all the mods except for LLib, Wings and Backpacked - fine work confirmed. Deleted Backpacked, added back everything else - flight still worked. Added Backpacked - flight disappeared. Weird...
Anyway, sorry for the false alarm.

debug#1.log
debug#2.log
debug#3.log
debug#4.log
№1 - the whole modpack
№2 - one of this bunch causes the trouble
№3 - it's Optifine
№4 - but when I remove Backpacked, things work fine

I've tested a couple times more. First, it's Optifine & Backpacked that cause the bug. Second, the bug itself appears to be somewhat else.
I can't fly/see the wings when I drag and drop them in the armor slot, but i can if I press RMB holding them.
The thing is, they can't be removed, because the armor slot is empty. For real - I can put on a chestplate and still fly.

The wings are only removed when I reload the world, die or travel to another dimension (wearing them out might work too).

I wasn't able to reproduce with issue with Forge 14.23.5.2847, Wings 1.1.6, LLibrary 1.7.19, Backpacked 1.4.2, and OptiFine F5. Also the right clicking likely equiped the wings in the baubles slot which let you have an empty armor slot/wear armor.

I do see your log loading an old version of LLibrary coremod (1.0.11), could you delete the "/mods/memory_repo/" directory and test again?

It must be Backpacked in combination with another mod then, if I could get your debug.log it may help figure out which.

Okay turns out I was mistaken with coremod versioning, it actually is independent of llibrary version. The last difference I see is you using java 1.8.0_251 while I used 1.8.0_51 which is what the vanilla launcher comes with.

debug#5.log
Done, but it's the same coremod version again.
Checked the jar in the mods folder, it contains 1.0.11 coremod.
Downloaded latest LLibrary from Curseforge, 1.0.11 inside.
Should I update from another source?
P.S. misclicked on Close+Comment

Updated Java to 1.8.0_261 and tested.
Wings still go in the baubles slot if RMB pressed, however drag-and-dropping in the armor slot works too.
IDK, the problem is sorta solved for me, since I don't use Baubles, but in the end it's still a bug, most likely caused by pre-release Optifine.