I won't get into what +John Chrysostomos was sermonizing against save to say it wasn't the Jews, it was the Judaizers which I suspect in his times meant the messianic jews based on what you have said.
That was in fact what set him off: Apparently, quite a few Gentile Christians were joining their Jewish brethren in keeping God's Appointed Times. Since by this point the definition of "Judaizing" had morphed from the Biblical definition of "formally undergoing the ritual of circumcision to become a Jew in order to be saved" to "keeping the Torah"--which would of course make the Messiah and all of the Apostles Judaizers and heretics by the fourth-century Church's standards.
Therefore, the so-called "Golden-Mouthed" wrote his eight Homilies Against the Jews, which you can read here if you'd like. We can see his purpose in his opening homily:
(I.4) . . . Another very serious illness calls for any cure my words can bring, an illness which has become implanted in the body of the Church. We must first root this ailment out and then take thought for matters outside; we must first cure our own and then be concerned for others who are strangers.Interesting, isn't it, that the mere thought of Christians keeping the same Feasts that their Lord kept, doubtless wanting to know more about Him and the way He lived, set him off so?(5) What is this disease? The festivals of the pitiful and miserable Jews are soon to march upon us one after the other and in quick succession: the feast of Trumpets, the feast of Tabernacles, the fasts. There are many in our ranks who say they think as we do. Yet some of these are going to watch the festivals and others will join the Jews in keeping their feasts and observing their fasts. I wish to drive this perverse custom from the Church right now.
Read his work. Isn't it evident that in his mind the very concept of "Messianic Jew" is an oxymoron? It's not surprising. In his day the Church had adopted a stance of reconciliation towards the Roman Empire and hostility towards the Jewish people--even those who were believers in the Messiah Yeshua.
He calls the Jews demons (Homily I.vi.3), and repeatedly condemns them as the slayers of Christ--it apparently misses his attention that Yeshua Himself said, "No one takes [My life] from Me, but I lay it down of Myself" (John 10:18) and "Father, forgive them, for they do not know what they do" (Luke 23:34). He takes that which God commanded them to do, like the daily sacrifices, and twists it into a sin upon them. He acts as if those in the Bible who were righteous weren't Jews themselves.
In other words, he twists the Scriptures into an anti-Semitic parody of themselves. So I have no respect for him, and it saps my respect for the fourth-century church that they dubbed such a twit "the Golden-Mouthed."
"Read his work. Isn't it evident that in his mind the very concept of "Messianic Jew" is an oxymoron?"
Indeed I have read his work; everything extant several times over. Christian Orthodoxy's roots are deep in Judaism, but we are not Jews of any kind and our covenant with God is not the covenant God made with the Jews. And yes, I do think to his mind the idea of a Messianic Jew is an oxymoron, but I believe he felt this way because to him if one believed that Christ was the Messiah, if one could sincerely pray the Creed, then one was not a Jew but rather a Christian. If one claimed to believe in the Messiah/Christ and could not recite the Creed and accept the The Church as the only sure venue of theosis, one was a heretic and heresy destroyed and destroys souls. He and the other Fathers, both before and after him, were equally outspoken about other heretics. By the way, I agree with him at least to some extent.
"So I have no respect for him, and it saps my respect for the fourth-century church that they dubbed such a twit "the Golden-Mouthed."
He is my patron saint. I celebrate my name day on his feast day and his icons hang in my home and office. I doubt we could be further apart in our estimations of him.