You are coming at all of this from center-left angle so your conclusions don’t necessarily fit with the conservative narrative.
But think about other issues near and dear to conservatives’ hearts and minds. At least to some degree, many of them can be framed as privacy issues.
You can’t frame them as privacy issues with people that don’t recognize them as such. Their view is that there is no right to privacy. It’s a made up right and there is no need for it to exist.
Gun Ownership. Certainly the main argument here rests on the Second Amendment to the Constitution in the Bill of Rights. But if you look past the basic right to bear arms, matters of privacy begin to be involved.
Take gun registration. Many gun owners fear that registration of firearms is a prelude to confiscation of guns at some future date. Opposition to gun registration can be seen as a right to privacy in that context — gun owners want to have privacy regarding the number and kinds of firearms they own, and they believe the government has no business knowing that information.
You get to the same conclusion they have but take an entirely different route.
If you read Libertarian legal theory, this goes back to English Common Law, philosophical theories of John Locke, etc.. prior to the existence of the Constitution. Gun ownership and “privacy” is rooted in property and contract rights, not a right to privacy. What you see as privacy issues, they already see as protected under the 4th Amendment’s right to be secure in their property and possessions, the 5th’s Amendment’s right against self-incrimination and the 9th Amendment’s limit on government’s ability to to deny or disparage rights retained by the people.
The right to privacy isn’t listed in the Constitution. The 2nd, 4th, 5th and 9th Amendments are.
While people who accept that there is a right of privacy would logically see gun registration as a privacy issue, people who don’t believe in a right of privacy still see registration as government interfering with their right to be secure in their property/effects and as a potential setup for self-incrimination should possession of those guns be outlawed.
Property Rights. Land owners, particularly in the Western states, feel they are entitled to make their own decisions about land use privately, without government interference and regulation. Water use, mineral rights, livestock conditions, and other factors, they feel, should be up to the individual farmer or rancher. In these days of drought, for example, why should anyone else get a say in how much water (that exists on his or her own land) the family farmer should be able to use? Who has the right to put restrictions on that and other practices? Aren’t those private decisions?
This goes back to the same issue as gun rights. Land ownership is a property right. Under Libertarian theory, what is on my property is mine. The government can’t come on my property nor can they prevent my from doing what I want with my property in advance. If a farmer does something on their property that somehow degrades the property of a neighbor, that’s an issue between them to be settled by them or, if need be, in a civil court. It isn’t up to the government to attempt to control their property preemptively.
Medical Decisions. Leaving abortion aside for the moment, conservatives had major problems with “Obamacare” (aka the Affordable Care Act) because they believed that the government should not come between a patient and his or her physician. Of particular concern were the so-called “death panels,” which, if any had been implemented, might have led to government personnel having a say in “when to pull the plug on grandma,” or whether a disabled child was ever going to be a “productive citizen.”
Surely end-of-life decisions and those regarding the amount of treatment a person receives are sacrosanct, the ultimate in discussions that should occur privately between physician and patient.
This and the abortion issue are tied directly together. Once again, under Libertarian legal theory, there is no “privacy” in either situation. “Privacy” is an entirely individual issue. Once you involve a 2nd person, privacy no longer exists. If you wish to keep something private, you keep it to yourself.
For the Libertarian, any time something happens between two or more people, that falls under contract law theory. When you make an appointment with your doctor for medical care you are establishing a contract. Within that contract you two can agree to keep any information revealed as “confidential” but there is no automatic right to privacy. It is an inter-personal agreement where the people involved decide how much to reveal to each other based on whatever level of trust that have established between them. If your medical provider breaks that trust and reveals information they had previously agreed to keep confidential, it is a breach of contract and you can sue for damages (defamation?). Their breech would also free you of any requirement to keep their information confidential.
The Obamacare/Death panel nonsense was also seen as government interference with the right to contract. If government establishes “death panels” to parse out health care and decide if doctors are or aren’t permitted to provide care to someone who requests it, that is government interference with their right to contract.
This is where there is a argument for legal abortion, marriage equality, etc..
Government regulation (or prohibition) of abortion and marriage interferes with the individual’s right to contract. This is why most Libertarians aren’t against abortion or LGBT rights. (If I recall correctly their 2016 Party platform essentially said “These are none of our business”.) But if you want to argue those with a Libertarian, use “right to contract” instead of “right of privacy” and you’ll be on the same page.
The social conservatives tend to fall in the middle and pick and choose between the Libertarian mindset and there being a full right to privacy so their views are much more complicated and conflicted.