I thought that I heard that the American artist who drew the chapter pictures and cover and such read the book before she did the artwork, but the British artist did not. So if the British artist didn't read the book, then there is no was that they could have known about who was in the order or anything that went on in the book. They probably just thought "Hmm, Order of the Phoenix... Order of the Phoenix... Order of the PHOENIX! That's it! It's so simple! I'll draw a Phoenix!"
Oh about the bowtruckle tree, good job on catching that, I didn't even notice
But even if Harry broke off a teig, he couldn't use it could he? There's not a dragon heartstring or a phoenix feather in a twig, no magical core at all. Maybe there would be a little sap,
, but nothing magical