Educated Guesswork

I think likely in the 19th century, "doctor" was basically shorthand for "opium derivative supplier in elixir form". Yes, he provided not much in the way of get-you-better-faster care, but he probably did keep some pretty good analgesics around. Without the doctor, you're probably more or less limited to alcohol for pain management.

Bullet removal and bone setting are both pretty important. If the bullet isn't removed that guy would have died slowly and very very painfully from lead poisoning. If bones heal wrong he could be partially crippled. That's not to say that medcare in the mid 19th century wasn't primitive compared today, for example they still used leaches then. But it was still important to have medical procedures performed, by a doctor or some other trained person then as it is today.

Note: the link is bad; it says
    http://www.us.imdb.com/title/tt031635\
instead of
    http://www.us.imdb.com/title/tt0316356

People have always had near-mystical beliefs in doctors. Most people would be very surprised to learn to what extent that is still operating today.

Life expectancy over the 20th century increased by 30 years in industrialized countries, yet the consensus among health care economists is that medical interventions in the usual sense (immunizations, doctor visits, hospitalization) can account for only a small fraction of this total, perhaps about 5 years' worth. See, for example, "Improving Health: Measuring Effects of Medical Care", by Bunker, Frazier and Mosteller; Milbank Quarterly, 72(2), 1994.

It's likely that to a large extent, medicine today does as much harm as good, just as in the 19th century. But people are amazingly resistant to this conclusion. They prefer to retain their belief in the magical powers of doctors to make them well, just like in the old days.

I think there are a couple points here:

a. Treating deadly childhood illnesses (mostly crapping yourself to death) is the source of the big mortality wins, and you can do most of that with the germ theory of disease and Roman civil engineering technology--good water supplies, good sewers, managing trash properly, washing your hands before preparing food and after treating sick people, etc. Add in rehydration therapy (boiled water with the right amount of salt and sugar, basically) and you've taken care of a huge amount of infant mortality. By contrast, a lot of the medical miracles of the last 30 year are either about treating rare conditions (your chances of surviving childhood cancer are much better) or about extending the life of older people (bypass surgery is a good example).

b. A lot of the improvements in treatment make things less miserable or more convenient, rather than solving some problem that would otherwise have killed you. Most people on antidepressants, proton pump inhibitors, better NSAIDs, etc., aren't having their lives saved by the better treatment, they're just having their quality of life go up. That's worth something all by itself.