1. The baby is affected by the epidural, which can result in several days of grogginess, increased difficulty in establishing breast feeding, etc.
2. There is a natural amnesia for the pain of childbirth. If you get an epidural that effect gets dropped. So there may be less pain, but the woman tends to remember it more clearly.
3. An epidural greatly limits the mother's range of possible positions and activities. For example squatting in a shower can be great for progress, but you can't do it if your legs don't work properly.
4. One intervention leads to another. An epidural slows the birthing process, and massively increases your likelihood of further interventions. In the linked article, the woman wound up with a C-section. The odds of that happening would have been considerably less if there had not been an epidural. (Even so, if there is no fetal distress, why the hurry to have a C-section if that had not been discussed? Before the childbirth they should have had a birthing plan, and the doula should have known the mother's wishes and defended them. That's what she's there for...)
That said, natural childbirth is incredibly difficult and not always possible. If a woman wants and is capable of having a natural childbirth, my hat is off to her. But women shouldn't be pushed to do that if they don't want to.