I think you're coming from a different perspective. You care whether a quantity is easy to measure and whether it's a good predictor of something that's useful in practice. I don't care very much about that, because in pre-paradigmatic fields it's too early to ask for practical applications anyway. Instead I care whether the quantity can serve as a good building block for future research, and for that it needs to be a hard number rather than a soft one, so to speak. (Maybe I should've called them hard vs soft, instead of true vs fake.)
Whether something is easy to measure matters a lot whether it's a good building block for future research. If something is easy to measure and it's a good predictor of other qualities it provides a good building block for further research.
Easy to measure means that you can do research and study how the variable interacts with other variables. That's the core of research.
That means you care whether the measurement has random and systemic noise but you don't have to ask for realness.
Do you know more degrees of freedom of a system through having a measureme...
If you believe that science is about describing things mathematically, you can fall into a strange sort of trap where you come up with some numerical quantity, discover interesting facts about it, use it to analyze real-world situations - but never actually get around to measuring it. I call such things "theoretical quantities" or "fake numbers", as opposed to "measurable quantities" or "true numbers".
An example of a "true number" is mass. We can measure the mass of a person or a car, and we use these values in engineering all the time. An example of a "fake number" is utility. I've never seen a concrete utility value used anywhere, though I always hear about nice mathematical laws that it must obey.
The difference is not just about units of measurement. In economics you can see fake numbers happily coexisting with true numbers using the same units. Price is a true number measured in dollars, and you see concrete values and graphs everywhere. "Consumer surplus" is also measured in dollars, but good luck calculating the consumer surplus of a single cheeseburger, never mind drawing a graph of aggregate consumer surplus for the US! If you ask five economists to calculate it, you'll get five different indirect estimates, and it's not obvious that there's a true number to be measured in the first place.
Another example of a fake number is "complexity" or "maintainability" in software engineering. Sure, people have proposed different methods of measuring it. But if they were measuring a true number, I'd expect them to agree to the 3rd decimal place, which they don't :-) The existence of multiple measuring methods that give the same result is one of the differences between a true number and a fake one. Another sign is what happens when two of these methods disagree: do people say that they're both equally valid, or do they insist that one must be wrong and try to find the error?
It's certainly possible to improve something without measuring it. You can learn to play the piano pretty well without quantifying your progress. But we should probably try harder to find measurable components of "intelligence", "rationality", "productivity" and other such things, because we'd be better at improving them if we had true numbers in our hands.