I agree with and like William Zinsser on many points, and I believe he has helped countless budding authors and language learners over the years to develop good writing skills. I know I read his books years ago and profited from them (I will never forget a funny introduction to
On Writing Well, where he highlighted the point that good writing skills can only be developed through persistent, hard work over a long period by featuring a conversion between an author and a brain surgeon - who did writing "on the side" - in which the author chimed in that he too planned to do some brain surgery "on the side").
That said, English is a rich, beautiful, and varied language and we should be careful not to box ourselves in with arbitrary rules that limit our style. Mr. Zinsser's admonition not to use or over-use Latin-origin words does just that and is therefore too arbitrary to be productive.
Take the word "productive," which I just used. At its root is the Latin duco, ducere, duxi, ductus, generally meaning "to lead." So we get, in English, a rich range of words such as duke, duct, deduce, reduce, induce, introduce, produce, and all their variants (deductive, inductive, productive, etc.), all useful words in the right context.
Now there are countless words like this in English, and I don't think it is meaningful to try to sort them out from words having an Anglo-Saxon lineage and others thought to have punchier qualities. After all, well over half of English vocabulary derives from Latin and it seems self-defeating to limit yourself to less than half of the words available to you in choosing ways to express yourself.
Yes, do avoid writing in a bloated or hopelessly abstract style. Use punchy verbs to liven things up as occasion allows. But don't waste your time trying to sort words out by their etymological origin because words derived from Latin are not "the enemy" and, even if some of them are stilted, many or even most are not.
If I may suggest my own rule: do not mechanically apply arbitrary rules on word choice but rather choose your words to suit the context, following your instincts for what sounds right for your intended reader.