I think there is a significant difference between being irreplaceable because of skills you have (ie, Einstein or Da Vinci were and still are irreplaceable) and being irreplaceable because you artificially make you so (as Anne does in your example). The first one may be undesirable, but the only way to overcome it would be to just not have genius, and I don't think it would do good to humanity. Einstein or Da Vinci shouldn't have limited themselves so they would stay replaceable.

The second one (the Anne of your example) is definitely a problem, a subset of the more general case of people doing bad work for their own personal benefit.