That murder analogy is claptrap, and I suspect you know it.
First, a murder has nothing to do with the privacy of medical information, which was the grounds for a US constitutional prohibition on banning abortion. Second, any right to privacy of the murderer is pretty obviously trumped by the pre-existing right to life of the person being murdered. (This right-to-life argument, incidentally, is the one with which anti-abortion campaigners have been most successful outside of the US: if you can secure a legal ruling that an embryo has equal rights to a born child or adult, then suddenly you can argue that early-term abortion is homicide. This is fine example of making an argument from a faulty premise, but that’s kind of par for the course from a lobby that has a large-crossover with young-earth creationism.)
As it happens, I live in a country with legalised abortion, and judging by your username, I suspect you do too, or at least did (few Americans would understand the reference in your username). And yes, I did vote for it, directly. But unlike Americans, I live in an actual democracy. And it’s one with an electoral system that hammers extremists, and while we have a written constitution, it is one that is routinely amended by public vote. Yes, that means we have put some dumb things in our basic law over the years, but we also get to emphatically remove dumb things from our basic law too.
I agree with you that the US places too much emphasis on its constitution, especially now that the practice of amending that constitution to respond to changing events appears to have died off. The last amendment was in 1971 - over half a century ago (and all it did was reduce the voting age to 18). In the 20th century, the US Constitution was amended 11 times (or 9 if you discard Prohibition and its repeal). What used to be a living document of basic law is at risk of being ossified into a snapshot of early-20th century beliefs.
But there is nothing in the US constitution that actually prohibits abortion - it’s just that the Constitution is one of the few legal instruments that the conservative Right are wary of interfering with. Many US states have legal abortion simply because they passed laws that allow it (this is the same as the UK). However, in many other states, religious conservatives, imposing the morality of their particular sect upon the entire population, retained or strengthened laws that banned the practice. In the US legal system, the only way to counter repressive laws at the State level is to show that they are incompatible with the Constitution. This used to work, but Roe v. Wade was both a victory and the start of a defeat for the progressive faction in the US. But Roe v. Wade didn’t assert the right to bodily autonomy explicitly, all it did was assert a very specific right, but one that was incompatible with any attempt to criminalise abortion. I agree that this is an arse-backwards way of doing things, but if you spend any time in the US (and outside of the cities), you’ll understand that this is a country that even today is deeply religious, and very culturally conservative, and we’re talking about 50 years before today’s more liberal outlook: after all, much more “liberal” countries in Europe, taking a direct human rights argument that women, as human beings, have the right to full bodily autonomy took longer to legalise abortion.
I don’t disagree for a moment that “privacy” is a weak hook to hang the right to abortion upon; but it was the only hook that could be used in 1971, given the absence of an explicit right to bodily autonomy in the US Constitution - the US Bill of Rights does not have that concept, partly due to it being written in 1791, and partly due to the existence of chattel slavery across the USA at this time. That need to use a roundabout way of asserting a human right is why the current mess exists. (The 1948 Universal Declaration of Human Rights was ratified by the USA, but as a declaration, it has no legal force)
So that was the victory, but the beginning of defeat was that Roe v Wade spooked the Religious Right so badly that they formulated a long-term plan to load the US Supreme Court with the most right-wing pro-religious candidates they could find over the coming years: this way they could neutralise the parts of the Constitution that don’t agree with their ideology without having to open the Pandora’s Box of actually amending it, and it paid off last year, when a judgement came down to say that no, there’s no right to medical privacy, so no, there’s nothing to stop States passing laws that make abortion a crime.
Oddly, you seem to be labouring under the impression that the USA is a functioning, representative democracy. I would disagree, given how many barriers there are to ensuring that the choices of the electorate are reflected in their government. State-level gerrymandering (sorry “redistricting”), voter suppression (sorry, “anti-fraud measures”), even the Electoral College mechanism are all designed to prevent the actual voice of the electorate being heard. At least the Electoral College may have made sense in the days of pony post, but it has no purpose now; the other two are modern tactics to lock-in supermajority results, often from a minority voting base.