This is going to be reeeeeally hard for you when they're using five moral intuitions and you're using one (care/harm).
For instance, suppose someone says, "But life is sacred!" (purity axis). If you go, "Blah whatever, silly words. Appeal to goodness," you totally miss the point. Part of goodness (to them) is purity. It's preserving and respecting the sacred.
If someone says, "We have to allow people who have arthritic hands to own slaves so they don't have to do their own dishes," and you argue for it by talking about how uncomfortable it is to wash dishes with arthritic hands, you're going to get nowhere. Because people are going to, quite, rightly, say: SLAVERY IS EVIL! In no way does the moral call to reduce arthritic pain override the moral call to not enslave people.
So if you're going to reach people's emotions, you have to understand the emotions that they have. And if they have the "life is sacred" emotion--and a heck of a lot of them say they do--bringing up a lesser moral concern ("it sure is burdensome to carry and then probably raise a child") is not going to cut it.
Also, Medium is absolutely full of stories appealing to emotion. There are only a handful that deal with primarily fact-based arguments.
So I think you're precisely 100% wrong. This kind of emotional appeal is already ubiquitously present for everyone to see, has been for decades, and most people who could be persuaded by it already have been. And it's enough to get pro-choice legislation passed! As long as people consider it their top issue!
If you want more support, then you need to tackle the foundations of the moral intuitions.
And that does take facts, and that can work, though it is tricky, because if you're the rabid frothing enemy opposition, everything you say is a trick and you won't be listened to.
But get the facts in place first...or, if you can, find a way to appeal to all the moral intuitions that the target is likely to be using to make their decision.
(Reference for moral intuitions: The Righteous Mind, Haidt, 2012.)