From the paper:
… the RR when looked at from the diagnostic testing angle, is now understood to be a likelihood ratio and therefore a ratio of two odds not … simply considered as a ratio of two conditional probabilities…
I have no idea why you are converting proportions to odds ratios here. This flies in the face of decades of mathematical work on the statistical properties of estimators and experiments.
In an experimental design context, the only thing we have control over is the error probabilities, which are more informative than likelihoods.
After the data are collected, an observed proportion can be useful to estimate a future probability.
Pawitan, paraphrasing Fisher wrote:
whenever possible to get exact results, we should base inference on probability statements, otherwise they should be based on the likelihood.
Pawitan, Y. (2001). In all likelihood: statistical modelling and inference using likelihood. Oxford University Press. p. 15
The Bayes Factor has a frequentist interpretation of \frac{1- \bar{\beta}}{\alpha} , which is a ratio of probabilities.
https://www.sciencedirect.com/science/article/pii/S002224961600002X
Diagnostic testing is merely a specific case of the broader hypothesis testing problem, where:
Type I error = 1 - spec
Type II error = 1 - sens
Power = sens = 1 - \beta
Information is formalized as a (pseudo) distance metric of 2 probability distributions.