Two issues here:
First - is Warbler a taxonomically monophyletic definition, or does it refer to a particular morfotype / ecotype of small bird? In some cases, I think there is value in keeping some groups monophyletic. IE, if a bird turns out to not be a Cotinga, or Babbler, or Antbird, perhaps worth preserving the meaning of those. However Tanager, Warbler, Flycatcher I think are pretty broadly understood to mean a certain shape and behavior more or less rather than a monophyletic grouping.
Second - those birds are called Warblers entirely due to the Brits who colonized the rest of the world and applied their names to the birds they and their progeny encountered along the way. So at some point, this logic of "my version of English is correct, Warblers are old-world, American Robin isn't a Robin, etc" is sort of comical and I'm not sure where it gets you but whatever.