I remember that I have seem many places (e.g. OG10-233) that the subject and the "to be" in the while clause can been omitted and the verb in the while clause becomes a present participle.

However, OG considers the following sentence grammatically incomplete (yes, I know that I am not supposed to post any OG question in this forum, but hopefully not posting the whole question won't violate the rule here ;-)):
Many people have a disinclination to recognize the weakness of their analytical skills while willing to admit their lack of computer skills or other technical skills.

Then, I saw that nytimes.com has an article contains a sentence with a similar structure:
While willing to adjust any individual cases of injustice or inequality within the present agreement, representatives of the anthracite mine operators today notified the mine workers that they were against granting any general additional wage increase or taking any action that would tend to reopen the award of the United States Anthracite Coal Commission.

Can I conclude that if I want to OG consider "while verb-ing..., main clause" correct but "while adjective..., main clause" incorrect?

Thanks in advance.

S+V+O+present participle+pronoun (referring to a noun between V and O in the main clause) while present participle...

Above structure is from OG10-178.

How come such structure is permitted? I thought that the omitted subject in the while clause has to be the subject in the main clause.

the first, and most important, thing i can say here is that if the o.g. disagrees with the new york times, then you know who wins the argument. (hint: it's not the new york times.)

with that said:
there are two very distinct uses of 'while'.

1) 'while' can be used as a subordinating conjunction, to indicate contrast.
if it is used in this sort of way, it MUST be used with another CLAUSE (i.e., not just a participle).
for instance:
while i had anticipated a disturbance, i had no idea it would be so big. --> correct
while anticipating a disturbance, i had no idea it would be so big. --> incorrect

2) 'while' can be used to express simultaneity (i.e., X happened during Y).
if there is no change of subject, then the preferred construction is a participle.
while sleeping on the train, i dreamt of a seven-foot-tall fairy goddess --> preferred
while i slept on the train, i dreamt of a seven-foot-tall fairy goddess --> not preferred, although probably still ok