I think I've over simplified the bird method but basically they inhale and exhale at the same time as required. They literally have double ended lungs, you push air in at one end and exhale CO2 at the other end. We mammals use the same route in and out of our lungs and the whole thing is driven by our diaphragm which pumps the bottom of our pleural cavity.
The bird mechanism is obviously efficient for oxygenation but it must have a cost that our body plan discarded or at least failed to even consider many millennia ago.
> ... it must have a cost that our body plan discarded ...
Our body plan didn't discard this mechanism, any more than it discarded wings or beans. We never had any of those features in our ancestry, because mammals aren't descended from birds. Our most recent common ancestor is much, much earlier than bird ancestors began evolving any mechanisms related to flight or high-altitude breathing.
Is this 'just' continuously blowing out from air stored in cheeks, while breathing as normal through the nose? I have said 'just', but it doesn't seem easy!
It doesn't seem to me that the lungs are being used differently (to normal breathing). I could be wrong.
(Edited to add text in parenthesis in 3rd sentence)
Yes, take a glass of water and fill your mouth, pretend to be a fountain spitting a stream of water, and breath through your nose. That is how I was tought to learn it anyways.
The bird mechanism is obviously efficient for oxygenation but it must have a cost that our body plan discarded or at least failed to even consider many millennia ago.