Since the tags can really only be used once, and I am assuming the cardboard is specifically for writing ease/clarity. I would think once the cardboard has disintegrated, the tag remaining will be fine.
I am going to put an order in for them also. I think they look great! I loved the washer idea too! I would use something like them - getting some cheaply - to use in the medium when repotting to ensure a tag loss never means a noid plant.