Law actually seems quite similar to programming. You can think of the jury as your users and facts/precedent/laws as the statements available in your programming language. Then your job is to assemble the statements into a program that compiles and that your users want to buy.
why do so many programmers think their job is inherently more special and difficult than everyone else's?
Probably largely because they get paid more than all of their peers when they start their careers.
But potentially life-or-death cases are not limited to death penalty cases, they are just the most obvious class of cases that are life-or-death.
There are far greater numbers (and proportion) of lawyers who work on potentially life-and-death work -- not limited to death penalty cases -- than programmers who do so.
Though for what it's worth this is less a "programmers are awesome and important!" argument than it is a "don't get your view of lawyers from television shows" argument -- if the OP had gone with doctors or something I wouldn't have said anything.