The Church is unable to remove the needs - they're hardwired - but it doesn't really need to. All it needs to do is make people feel bad about having them and acting on them.
People who feel bad about themselves express that through masochism, or though authoritarian and abusive sadism - which is exactly the kind of acting out we've seen in Ireland, in Germany, in the US, and in Africa.
Sexual morality is not the issue, and never has been. The issue is power through psychological manipulation.
The Church discovered - or rather reinvented - the use of sexual and other ethical double-binds for mass political psychological control. But it was only able to enforce its brand of religious totalitarianism while it had exclusive control over the narrative space.
The fact that monogamy is waning and birth control is increasing are proof that the Church has lost its narrative monopoly - not that it has given it up voluntarily.
People mostly don't understand the distinction between process and content, which is why 'religious' leaders find manipulation so straightforward.
The Church lost its hold in the West because it was pushed out by competing narratives, not because the processes it used to try to legitimise its narratives were ever fully deconstructed. That was effective as far as it went, but individuals remain vulnerable.
Disagreeing with religion, Dawkins style, isn't enough, because content and belief are a cover story for psychological process, and arguing with the content isn't a very effective form of attack.
The next stage of secularisation will be wider awareness of process. Once that starts happening - and it's beginning already in other areas - social and political institutions that derive their power from psychological manipulation will have a tough time surviving.
It's going to be a very interesting century.
However, you're still assuming, a priori, that the Church is the principal and not the agent in this story. Your narrative puts the Church in the role of being the one trying to manipulate repression and psychological conflicts for the ends of power, when it seems just as likely that the Church is merely the tool of other political actors toward that end. In fact, the main scholarship of the left on this topic, in the works of people such as Foucault and Hannah Arendt among others, would put the church in the role of victim/tool of totalitarian tendencies of the modern capitalist, law-based state toward the ends you describe.
With your rhetorical skills, you can derive why you're wrong about the entire direction of your objections. Align culture with our nature.