Why is only "artificial" birth control supposedly bad? Why is the narrative that "life begins at conception" advanced? Neigher has any logical basis.

I think there's a bit of a misunderstanding on the first point. Prohibitions on artificial birth control aren't necessarily premised on the idea that an unfertilized egg or sperm are human lives. At least in Catholic Christianity, they are considered to violate moral precepts. Specifically, religious beliefs that one ineluctable purpose of sex is to procreate - so that engaging in sex while deliberately preventing the possibility of conception is a moral violation. It violates what they believe is G-d's command to "be fruitful and multiply."

As for the latter, I don't know if this is a "logical" basis or not, but I think it's based in part on the idea that there has to be a specific defined moment when human existence begins. And it's much harder to articulate that defined moment as lying on a spectrum, rather than one of the two categorical changes in human reproduction: birth and conception. Both of those changes happen in a near instant, while other developmental events happen in a gradient. And since many (most?) people feel that an hour before full-term birth a fetus is actually a human baby, the fact that there's no single instant of category change before that will lead some (many?) people to search back to the last category change as when that switch from not-a-human to human baby happened.

It's a very human thing to be uncomfortable with the uncertainty that happens when change happens on a gradient - that's why many cultures have "rites of passage" where the culture deliberately defines a specific event/time that separates childhood from adulthood, and nearly all legal systems have the same thing. An arbitrary age of majority, where a person passes from childhood to adulthood in an instant rather than gradually. That's unsatisfying to some people in the context of when humanity first manifests, because at some level we recognize that "rites of passage" and ages of majority are inherently arbitrary - they exist because we need them, but they aren't necessarily the "right" moment when a child becomes an adult in any real sense.