For me, taking the logarithm makes it even more intuitive somehow - the test becomes something like a modifier in an RPG-style game. In your example, the test gives +2 to the possibility of having a disease, a more powerful test could be +3, etc. Adding all the different +/- modifiers works just as well, and all this math is easy to do in one’s head.
Jaynes was a big proponent of taking the logarithm as well (as you probably know), referring to it as “measuring in decibels”. Unfortunately, I don’t actually understand what this log p/(1-p) gadget (“logit”?) actually does, mathematically speaking: it looks tantalizingly similar to an entropy of something, but I don’t think it is? Relatedly, I don’t really see how this would work for more than two outcomes—or why it shouldn’t.
The logit function is just a mapping from non-log prob space to log odds space. It’s the odds formula wrapped in a natural logarithm. In one way, it’s not “doing” much of anything. It’s not the journey, it’s the destination.
Why hang out in log odds space? Well, it’s a bigger space. It’s [-inf, inf], which is bigger than the [0, inf] of odds and the [0, 1] of prob. Odds space means you can multiply without normalizing. Log space means you multiply by adding. And being a natural log means you’re good to go when you start doing calculus and the like.
You can also cover a huge range of probabilities in a small range of numbers. -21 to 21 covers 1 in a billion to 1 - (1 in a billion).
Edit: also, obligatory 3blue1brown video https://youtu.be/lG4VkPoG3ko