Don't forget all of Anti-Duehring being entirely a refutation of Duehring used as an opportunity to expound upon a variety of subjects. Using the preface to Socialism: Utopian and Scientific, which is part of that project:

As is well known, we Germans are of a terribly ponderous Gründlichkeit, [. . .] Whenever any one of us expounds what he considers a new doctrine, he has first to elaborate it into an all comprising system. He has to prove that both the first principles of logic and the fundamental laws of the universe had existed from all eternity for no other purpose than to ultimately lead to this newly discovered, crowning theory. And Dr. Dühring, in this respect, was quite up to the national mark. Nothing less than a complete System of Philosophy, mental, moral, natural, and historical, a complete System of Political Economy and Socialism ; and, finally, a Critical History of Political Economy—three big volumes in octavo,17 heavy extrinsically and intrinsically, three army corps of arguments mobilized against all previous philosophers and economists in general, and against Marx in particular—in fact, an attempt at a complete “revolution in science”— these were what I should have to tackle. I had to treat of all and every possible subject, from the concepts of time and space to bimetallism, from the eternity of matter and motion to the perishable nature of moral ideas; from Darwin’s natural selection to the education of youth in a future society. Anyhow, the systematic comprehensiveness of my opponent gave me the opportunity of developing, in opposition to him, and in a more connected form than had previously been done, the views held by Marx and myself on this great variety of subjects. And that was the principal reason which made me undertake this otherwise ungrateful task.

Day did not insist on having the conversation on Twitter that I had seen. Perhaps I missed where he did, in which case, would you mind pointing it out to me?

However, you will notice that Poverty of Philosophy was not, in fact, a letter written privately to Proudhon, but was published and distributed to the public so that the people who read Philosophy of Poverty could read Marx's rebuttal of it. Refuting a public figure in private is not a very useful practice.

To my knowledge, Day insisted only that the response be public and not on the specific medium where JT happened to respond initially, and if JT said he'd, I don't know, make a video or write an essay of his own or whatever, Day would accept that just as readily.