My favorite example is Stripe. They've never skimped on docs and you can tell they've made it a core competency requirement for their team.
At its core, a good test will take an example and do something with it to demonstrate an outcome.
That's exactly what how to docs do - often with the exact same examples.
Logically, they should be the same thing.
You just need a (non turing complete) language that is dual use - it generates docs and runs tests.
For example:
https://github.com/crdoconnor/strictyaml/blob/master/hitch/s...
And:
https://hitchdev.com/strictyaml/using/alpha/scalar/email-and...
You're right, it is a matter of culture and discipline. It's much harder to maintain a consistent and legible theory of a software component than it is to wing it with your 1-2 other teammates. Naming things is hard, especially when the names and their meanings eventually change.