I wasn't either. Yes, I used to use lip products, but I didn't have chapped lips.
Many people have chapped lips. Many people don't have chapped lips. Millions upon millions use chapsticks and other stuff on their lips, others don't. Some people with EC use or have used lip care products, others don't/haven't.
There is no proof that there is any connection between chapped lips and EC or chapsticks/lip products and EC. There might just be no connection at all. And we will probably never know the answer, so I'll stop speculating about it.