You got it right. I have never checked it out, just remember it and that it worked. And of googling it i get no hint that it could work, only that it is dangerous. Lots of different reasons for chapped lips .. https://my.clevelandclinic.org/health/diseases/22005-chapped-lips .
The story is true though, only wrong detail could be - could be - the acid, though almost certain it was dilute hydrochloric. I remember being told if not careful it would take enamel from teeth. Diluted, don't recall how much but it would have been plenty. It had a sour taste. And it worked. Like the - use needle to puncture athlete foot blisters then iodine to kill the fungal infection in the moisture - remedy. It worked too.
100% those were two home remedies dad used on problems drugstore solutions meant for the two problems did not fix.
It was Plato who said, “He, O men, is the wisest, who like Socrates, knows that his wisdom is in truth worth nothing”