To answer the OP in a nutshell: any invasive medical procedure carries with it some element of risk. If you read up the information on epidurals (there is a section on MN actually) you'll see that while it's very effective in blocking pain, it carries some risks, such as increasing the likelihood of instrumental delivery, which in turn, carries some risk to the baby. Unlike most situations, which affect only the person having the procedure, anything that you have done during pregnancy or labour is also going to affect the baby - so many mothers factor this into their thinking, when considering pain relief options, such as epidural, pethidine...
That's probably the most simple explanation of why many women plan to try to get through labour without certain types of drug.
Of course, the best laid plans can go wrong: I had to have a csection for one of mine, which meant having to have either a general anaesthetic, epidural or spinal. All of those carry some risk, but for that particular baby, a vaginal delivery carried a greater risk.
The comparison with taking a pill for a headache is a bit flawed, because if it's your headache, it only affect you. If that pill was going to affect the baby you were carrying, you'd probably view it slightly differently.
Having said that, many people would try to avoid medication for headaches, and would try other things to clear it first.
Bottom line is - it's weighing up the advantages and the disadvantages and coming to your own decision.
The cultural norms of different countries are interesting too - in some parts of the world it would be absolutely standard to be given an epidural, even with no medical need, whereas in other countries it would be unheard of. I find that fact in itself quite informative, ie: there is no 'set' level of pain which means that an individual 'needs' an epidural. A lot of it will be about the culture you live in, along with your own particular pain threshold, and your own views on the pain relief options.