Pick your tame up with an Argy and just drag it around to harvestable stone, you won’t have to do anything but watch it break rocks. Just stop at a stone while you’re carrying it. It’ll stop before your bird is encumbered. Rinse repeat.
On PS4 I tried to test it's obsidian gathering since it is "Rank 1", well all it was gathering was stone. So I went to another saying maybe something was wrong, NOPE it does not gather obsidian. Just use an Anklyo for that.