I'm a huge bird lover of the real world and Pokemon one, so probably bias here (haha!) but I think the idea is a good one in theory. I like the idea of Wind replacing Flying elemental type and Bird having it's own. So, Pidgey could be Bird/Wind whereas Swoobat would be Psychic/Wind and just get rid of Flying type altogether.
The whole "This Pokemon can fly but is not flying type" is a bit silly honestly. Several Pokemon can swim, that doesn't make them a water type. How odd would that be haha!