This issue seems to be with switching between regular item models and data-driven entity models, the latter of which all of these items use. A similar issue occurs when switching between an item and an empty hand.
This is an issue with the vanilla resources. It can be easily fixed by changing the layer of 'hotbar_parent' in the controls array of 'gui_hotbar_grid_item' to 5 in hud_screen.json.
'Pick block' does not keep a block entity's data by default using a keyboard, but rather when holding the 'control' key. The issue here is that neither touch nor gamepad controls have an equivalent to the 'control' key.
This has nothing to do with the shape of the bounding or collision boxes at all. This happens because these blocks use their bottom texture for particles, which have a lot of empty space.
On Java Edition, these blocks use the side and top textures for particles respectively.
Did they remove the ability to add biomes_client.json entries for biomes that don't normally have them? Or does this not work specifically with the Pale Garden? If neither of these are true, then it should still technically be possible to change these aspects of the Pale Garden.
Also, another biomes_client.json feature that is not mentioned here is "fog_ids_to_merge", which has been removed from the most current iteration of the file but still retains function.
This issue is not unique to chickens. All entities overlap their own shadows. It's just more noticeable on chickens since their feet are (effectively) a single plane.