Why would anyone believe that no bird today grows teeth?
Because today it requires tricks and mutations, whereas all archaeopteryx had them as a matter of course. Even creationists manage to learn that chickens have beaks and not teeth.
Moreover, the fact that it is at all possible to "trick" a chicken into growing teeth is another classic evidence of evolution. Chickens have teeth in their ancestry: it's written in their genes as well as in their fossil history. Stephen Jay Gould even titled one of his books Hen's Teeth and Horse's Toes.