Is this what "four-and-twenty blackbirds baked in a pie" refers to?
I think not in this case. Of course it doesn't make sense as it stands: people did eat blackbirds, but these couldn't have been baked and singing. But a dish with a blind-baked piecrust over live blackbirds would have been just the thing for a fancy showpiece at a nobleman's grand meal.
Huff paste pies were big ugly utilitarian objects, the medieval equivalent of catering-size food cans, and would not have appeared on a fashionable table. The contents would have been decanted and served more elegantly.