I think you guys make good points why the invisible plane should have a place in WW's mythos. Just give some kind of explanation and that's it. You are right, these superhero films shouldn't be taken so seriously. this is supposed to be mostly fun and escapism.
Sometimes I think Nolan did some real damage to the DC universe with his very 'realistic' and gritty take. It hurt Superman more than anyone. The only issue I find with the plane is that it makes no sense for Diana to have it if she can fly. Seriously.