Aside from the perennially annoying habit of putting the verb, "put it," after a long string identifying the source, I don't think that would bother me.
That is annoying. but in this case it doesn't bother me because what comes before "put it" is shorter than these things sometimes are.
On the one hand, it might not have been a whole sentence. Schiff might have said "The very least we know for sure -- and I tell you this with utter certainty -- is that the Russians mounted ..." So in that case "The Russians" was not the beginning of the sentence and the writer paraphrased the first part.
I'm not following you here. Surely you aren't suggesting paraphrasing something and passing it off as a direct quotation?
But I also don't think a sentence fragment quote requires "that." I would be fine with "At a minimum, some Trump campaign spending originated, as Schiff put it, 'with those zany Russians.'"
I completely disagree with you on the "that," but maybe that's just the way I was taught by various English composition teachers, including one who was also my journalism teacher.
On the other hand, I think that your rewrite is fine. It's perfectly grammatical and correctly uses lower case for the "w" in "with." "With those zany Russians" is just a phrase, not a whole sentence. "With" doesn't need to be capitalized.
Meanwhile, throwing a "that" in the sentence as is, as in your example, would strike me as wordy and unnecessary, even out of place. "At a minimum, Schiff put it that "the Russians mounted ..." I think it reads better without the "that."
I'm not following you here, either. You're saying it reads better without a "that," but you're putting in a "that."
I if you're going to take out the "that," then the sentence should read,"'At a minimum', as Schiff put it, 'the Russians mounted. ...'"
I note that Steve Coll is dean of the Columbia University School of Journalism, so he presumably knows what he's doing. However, I remain convinced that
"The" needs to be capitalized. That's my story, and I'm stickin' to it.
And I'm blaming
TNY rather than Coll.