There are a lot of species that just don't like being grabbed around the body like tiels, IRNs, bourkes. I saw a video where someone trained their IRN to lay on it's back and play dead so it is possible. It's just not my "thing" to train a bird to do what it's not naturally comfortable doing. But I do have all my birds grab-desensitized. I feel you never know when you're going to need to swoop down and grab a bird for its own safety.
So, just letting you know that it is possible with a bird that isn't naturally inclined to lay on its back.