Somewhere I once read that the diet of the modern western world is unusual in that we get most of our calcium from dairy products. In third-world countries today, and in our own countries formerly, people get/got most of their calcium by eating the bones of fish and small animals; that is, animals small enough that their bones, when cooked, could be easily chewed and swallowed along with the meat. I suppose blackbirds would be in that category.
The only instance that I can think of where this is done today is with very small fish, such as smelt or "sardines." Their bones are so small you hardly notice you're eating them.
I suppose the reason such things have disappeared from our diet is that cleaning them is so laborious, in proportion to the amount of meat. While I can imagine eating a blackbird bones and all, I suppose they did remove the feathers. Or did they?
From the above discussion, I infer that two kinds of blackbird pie existed: The pie from which live blackbirds flew out for the amusement of guests, and the more common kind, where blackbirds were actually cooked and eaten.
I like the idea of an entertaining faux-food centerpiece, and I hope the custom will be revived. The nearest equivalent today is probably the ice sculptures that chefs sometimes make, and the gingerbread houses that sometimes appear at Christmas time.