If you remove those c-sections that are medically necessary or recommended ("known indicators for cesarean delivery" is the language used in the abstract), then you're left with that population that truly "chose" to have a c-section. The association is stronger in this population.
I'd want to tie this back to those demographics. What kind of people truly choose a c-section? Is there something about that demographic that correlates to obesity for other reasons? Are obese women more likely to be counseled towards c-sections? Again, why? It's unclear if they normalized for this.
Incidentally, I think your speculation is probably wrong. You assume that c-sections are (or are regarded as) the lower-pain option. You can have a painless "natural" birth with an epidural, and post-birth recovery is going to be "painful"--in different ways--with either path. In other words, don't think people choose c-section for pain management in the first place, which invalidates the rest of your logic chain.