The morality of a fair and just society is one which does not impose on our liberty by taking away any more rights than is necessary for it to function, and which imposes those restrictions equally on all. And then instead of our miserably solitary state, “Behold, how good and how pleasant it is for brethren to dwell together in unity!” as the psalmist puts it.
So that’s a moral theory behind live and let live. If I explained it right, you’ll see that from that angle, the morality of restricting what consenting citizens do in private is only just if applies equally to all citizens and aids the functioning of society.
The connection people who disagree with the Church always seem to miss though, is that the restrictions placed upon us ARE equal to all. Sex would be equally sinful for me, for example, whether I had sex with my girlfriend of 4 years or my best friend. It doesn’t matter how much I love them or what their gender is, if I’m not married to them and/or what I’m doing has no biological potential to produce a child, I have committed a sin if I knowingly chose to do anything sexual with them. That rule applies to all people. The only people permitted to have sex with each other are a husband and wife, because it is in the child’s best biological interests to be raised by the two humans whose DNA makes up their own.
From a secular standpoint, your explanation of live and let live is logically ideal, because the assumption in that case is that what two people do between themselves with no others involved bears no harm to anyone else. But things are far more interconnected than that, and are never that clinical in reality. The argument here is that it is not morally just to restrict the private actions of any particular group. The problem with sex fitting into that, however, is that, according to the Church, sexual intercourse is NOT an action that is restricted to only specific groups, but both a privilege and responsibility
entrusted to only one. That is why the Church doesn’t consider forbidding sex between unmarried couples to be an imposition placed upon them. Marriage is a vocation in the Church. It is not a right, it is not an eventuality, and it is not a venture that is undertaken lightly. It is a Sacrament. Just as not everyone is able to become a priest or a nun, not everyone is able to get married. The purpose of marriage is, at its core, the unification of a male and female to produce a new human and provide a stable environment for their upbringing. The main point of marriage is not so much the couple as their children and their collective part in the continuation of human life.
A single glance at statistics for divorce and domestic abuse will show you how many people would have probably been better off emotionally and financially never having gotten married in the first place. For the record, I am in no way saying any children they had should not have happened. Nor am I saying a person should stay in an abusive marriage. But too many people get married without really being ready for the commitment it involves, as I mentioned before, and they just end up hurting their kids in the end. No child wants to watch their parents fight, and no child wants to see their parents have to separate.
With the Church’s stance on marriage in mind, restricting sexual intercourse to married couples intending to raise a family together is
vitally necessary for society to function. Without children, society would cease to exist. Like any ideal model, however, society obviously does NOT function that way, because those kinds of restrictions are not really viably enforceable.
Long story short, if a restriction on private action is just only if applied equally and provides some benefit, from the Church’s perspective, restricting sex to married couples would fall under that criteria, as no one aside from a husband-wife pair intending to have children may have sex, providing children with both biological parents who have a genetic and biological imperative to care for and nurture their own offspring. In this scenario, foster parents would be (like in nature) exceptionally altruistic individuals who are caring for a child as their own, despite having no maternal/paternal genetic relationship with them.
That example is unrealistically utopian, but the Church’s disagreement with allowing any consenting couple to have sex in private stems from it’s position that sex is less of a right for all humans than it is a privilege and vocation restricted to a married couple. I feel like I wanted to add more to this, but it’s really late here, so I’ll try to remember tomorrow.