I think you are misunderstanding several things.
First off, grammatical correctness does not imply meaningfulness as shown by the classic example (and vice versa, almost trivially. We are using lots of grammatically incorrect sentences and understand each other in most cases).
is a grammatically correct but meaningless sentence. It is just the same as 病気を見る is (at least) weird. Generally there is nothing similar to the sick (= sick people) in English, if that's what you had in mind.
Regarding no-adjective, a grammatical category (such as na-adjective or no-adjective) is a stipulation, so there's no definite answer as to its 'existence'. It is just a matter of how useful it is.
Consider na-adjective instead, for which there was actually a controversy (I have no idea how much it's settled, but na-adjective is standardly taught in Japanese school grammar).
Now think about the phrase
For those who accept na-adjective, this phrase is ungrammatical because 有名 is na-adjective and の should be な here.
On the other hand, for those who don't accept na-adjective, it is a grammatical sentence (because 有名 is just a noun and there is nothing wrong with 'noun + の'), but just not idiomatic.
So, at least as far as the above phrase is concerned, the unacceptability of 有名の人 doesn't imply anything.
Likewise, thinking that 丸腰を見る is ungrammatical implies you are stipulating no-adjective. Not the other way round. (Or at least, claiming no-adjective just by this example is not well founded).
I don't look up anything about no-adjective and never heard of it, but I guess it is just for an explanation following the thought like below:
- Some nouns have adjective-like meaning (丸腰, 有名)
- 有名 is followed by な to connect to nouns, hence na-adjective.
- 丸腰 is followed by の, hence no-adjective.
There does seem to exist research on な/の difference in Japanese linguistics. I stumbled on this article which has the following chart.
They are not inverse proportional, but anyway semantically similar enough words require different particles な/の. Thus no-adjective was invented as opposed to na-adjective.
FYI. Regarding replaceability with である, the best I've found is 瀕死の重傷 in the sense that 瀕死 is adjectival enough but cannot be replaced by である (重傷は瀕死である is nonsense).