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Ranks represent subjective degrees of belief but they do not, like subjective probabilities, nicely map to relative frequencies. So compared to probabilities, ranks are indeed limited in this regard. For instance, you can learn probabilities from data but there's no obvious way to learn ranks from data.

But for subjective belief they're still useful. Consider the problem of diagnosing a system with components that fail in rare cases. However we have no idea about failure probabilities. We can then use ranks. A diagnosis for some observed behaviour would then be the least surprising (i.e., lowest ranked) failure state that explains the observed behaviour. This is also the reason for least-surprising-first execution: the most important prediction or hypothesis is the most likely one and thus the least surprising one. There are some concrete examples in the paper which demonstrate this.

I am currently thinking about combining probabilities with ranks so that you can reason about both kinds of uncertainty in the same model. This could be implemented using a programming language that supports both ranked choice statements and probabilistic choice statements.