Ehhh it's definitely nuanced but we certainly haven't been dramatically overlooking anything fundamental.
The prevailing wisdom has been that a fully developed cockatoo has roughly the intelligence of a 3 year old.
A 3 year old figuring out how to use a drinking fountain wouldn't be world-breaking science, and I don't think this is either.
We have proven that they don't understand language and can simply mimic sounds. I don't think it's as deep as you are hoping.
Could they have their own language though?
African Gray parrots have been proven to understand significant amounts of language.