I have a friend whose child came down with chicken pox two days after being admitted to hospital for complications related to another vaccine-preventable illness. (She had an ear infection which was not caught in good time, though she had visited doctors for it, and by the time they realized what was going on, the infection had gotten into her skull.) She spent six weeks in Toronto Sick Kids' Hospital, in isolation because of the risk of chicken pox to the other kids in that hospital, received several weeks of IV antibiotics after that, and was on oral antibiotics for a total of six months. Twenty years ago, she would almost certainly have died. As it was, the two vaccines she hadn't gotten (the only two she hadn't gotten) turned out to be the ones that threatened her life. Chicken pox on its own is rarely life-threatening for kids, unless they happen to have something else that makes the complication especially dangerous. The same is true of mumps, measles, and several others.
Then there's the issue of population infection. You're right that naturally developed immunity is usually better and more effective for the child, barring a situation like the one I described above. The problem is that anyone who hasn't had the disease, or hasn't had it for decades, or has a compromised immune system, is at increased risk of developing it again, or of developing a spinoff disease from the same virus. Ever seen someone suffering from a chronic case of shingles? I have. It was passed to that person by their grandchild, whose chicken pox were completely uncomplicated. It left the elderly person in itchy agony for months and contributed to the decline that led to her death.
Then there's the issue of measles. Pregnant women who have not been vaccinated are putting their babies at risk if they come in contact with the disease. If they have school-age children who are not vaccinated, they've just multiplied their risk of exposure by the number of children they have. Measles in utero causes brain damage that is not correctable. Aside from saving my daughter the high fever and annoyance of the rash, I'd rather protect the unborn children of the women around me from accidental exposure through my child.
A vaccination is not just about the child in question. It's about a societal level of protection. Back in 1929, a bunch of my grandmother's cousins went to a Sunday School picnic. She didn't go - her baby sister was sick and she was needed at home. Of the forty-odd kids who went to that picnic, 30 were dead three weeks later, including most of her cousins, of diphtheria - a bacteria spread through stagnant water. Now, antibiotics would take a bite out of that number today - but the most dangerous time for diphtheria is in the first twenty-four hours, so many children would be in dire straits before their parents could get them to a doctor. Now consider polio. Fifty years ago, parents were lining up around the block to get an unproven, live vaccine for their children, knowing very well that the vaccine might kill them - and also knowing that contracting polio was a much bigger threat, because they had seen it and its effects. Polio has not been eradicated yet - it could come to North America at any time via an unvaccinated third-world immigrant, as tuberculosis has been doing over the last fifteen years. We have an illusion of safety due to the rarity of these diseases in our own lives, but it is an illusion. The smaller the world becomes, the more likely it is that one of these diseases will return to decimate society, and those who are unvaccinated will have contributed to spreading an epidemic that might otherwise have been contained.
I have already dug deeper. I've talked to doctors, to a friend of mine who is a microbiologist, I've read medical journals, I've read up on everything I can. Vaccines are the reason we as a society expect our children to get to adulthood. That has never been true before the last two generations, and it's still not true in many parts of the world. Good medical care in other areas is partially responsible as well - but the single biggest factor is mass vaccinations. As a responsible parent and citizen, I've decided that what is best for my children, my family, and my community, is to be protected to the furthest extent that medical science can provide.