Do you think it's possible to shift+left click armor to repair them up?

You should still be able to shift-right-click armor. It won't equip if it can repair

That's sadly not the case. It swaps just the armor.

Version information?

Latest on 1.21.

Fabric? Forge? Neo? And any other mods involved?

Fabric. Having a large mod list.

Can't reproduce with just fabric + fabric API + BTM, so it might be another mod on your list interfering

Thanks for letting me know. Maybe I look deeper into this and letting you know which mod is conflicting there.

You can still repair equipables. The thing is that when you shift click it though, it goes out of sync on the client and so your client thinks that you equipped it even though its still in your hand. If you just have another item (e.i. a chestplate and an elytra) that you can swap with, it should work, albeit very scuffed. (all my experience with this though is with it on the server and not on the client, so it might be different on singleplayer)