It's tied to the character, not the bird. I'm a capped digger no matter where I get my chocobo. And it's not unusual to not pull things up depending on your zone and skill level.
Generally, the higher level the monsters in the area, the higher skill you need to reliably have successful digs. Think of it as digging's version of craft tiers. You have to start in the beginner zones as you start out, before going into the higher level zones as your skill increases. Try digging in ronfaure/sartuabaruta/gustaberg and see if your results are better.
As an aside, you might try following around a goblin digger if you come across one. Every so often theyll stop and bury an item in the ground. There was a rumor going around that digging up what the goblin buries helps to raise skill. It seemed to push me to the next tier a few times when I was leveling years ago, but there's literally no way to tell for sure if it does anything, so grain of salt, etc.
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