I would say that it is the idiom "none the wiser" that you are thinking of. This means
to still be confused about something even after it has been explained
I would then surmise that the authors of the sources you have given have simply substituted other adjectives into this saying.
This Ngram shows that "none the wiser" has been used significantly more than "none the richer" or "none the healthier", but that "none the richer" actually predates "none the wiser" in print. I can't speak for whether "none the + comparative" was once a standard form, but certainly it only really caught on with "wiser".
Nonetheless, with reference to the common idiom, we can assume that these other uses mean "still not [something], even after [something]."
Your third quote, in context, supports this:
it is clear that the medical establishment was unable to help their
condition and after expensive tests and manifold prescriptions, the
individual returns home poorer and none the healthier
That is, they are still not healthy, despite having sought treatment at a medical establishment.
With an adjective that takes "more" in the comparative, rather than an -er ending, we would of course then have "none the more", eg, "none the more beautiful".
To address your queries: it is not the same meaning as "far from", and is nonstandard outside of the idiomatic "none the wiser"; as such, it is best used in informal contexts.