Have you read any of the criticism of the book? Its Wikipedia entry quotes Pullum:
> The book's toxic mix of purism, atavism, and personal eccentricity is not underpinned by a proper grounding in English grammar. It is often so misguided that the authors appear not to notice their own egregious flouting of its own rules ... It's sad. Several generations of college students learned their grammar from the uninformed bossiness of Strunk and White, and the result is a nation of educated people who know they feel vaguely anxious and insecure whenever they write however or than me or was or which, but can't tell you why.
(See also "50 Years of Stupid Grammar Advice" at, eg, https://www.cambridge.org/core/services/aop-file-manager/fil... . Language Log recommends https://www.google.com/search?hl=en&q=Strunk+site%3Ahttp%3A%... as a way to find further criticism from them.)
For "Omit needless words", a problem is identifying what "needless" means. Quoting http://itre.cis.upenn.edu/~myl/languagelog/archives/004552.h...:
> Whenever you see an appeal to ONW, you should wonder what people are doing with those "needless" words. Most of the time, those extra words are serving some function that conflicts with brevity; they're doing some work.
There are also examples how that phrase (likely) caused people to believe that if a word can be omitted then in must be ommitted, like http://itre.cis.upenn.edu/~myl/languagelog/archives/004176.h... and http://itre.cis.upenn.edu/~myl/languagelog/archives/000994.h... .
Or how in White's own writings he "sometimes throws in a few extra words just for the sheer resonant fun of it" - http://itre.cis.upenn.edu/~myl/languagelog/archives/001904.h... .
FWIW, I think "don't be too wordy" is more accurate and less likely to be misunderstood than "omit needless words", and one character shorter.
(Personally, I still recall my utter confusion in 11th grade English trying to figure what "passive" meant when told to avoid it. Turns out three of the four examples of passive in "The Elements of Style" ... aren't passive! And yes, I got marked off for using what may- or may-not have been the passive.)