Nobody is tearing apart the Russian church. It is quite strongly connected with the Russian state power levers anyway. And no ex-Jew(s) could do any serious harm to such a large institution even if they wished to.
Putin is having a bad dream or has decided to re-start Russian government antisemitism, or both.
But he’s right about his characterization of some people with Jewish-sounding names who have abandoned faith. There are a significant number of them who will answer “yes, I am Jewish” but have no idea what that means or requires (just as there are a significant number of folks who will answer “I am Christian” or “I am Catholic” or whatever but have no faith, either).
We are living in a time of strong secularization. Many have called this the “Post-Christian Age.” At least in USA and many European countries. Putin has apparently noticed a similar trend in his Russian church. From my perspective, it is a problem AND it is a problem that Putin seeks to blame some secularized Jews or ex-Jewish people for whatever is going wrong in his Church. Unfair to the Jewish people, obviously. Plus, by reverting to such ridiculous old rhetoric, Putin entirely misses out on anything he might possibly be able to do to help restore his wandering flock back to faith in God. Attacking or murdering some more Jews won’t help the Church or any Russians re-discover their church, faith tradition.
Zelensky banned the Orthodox Church even though it is the majority religion in that region.