"...but it's hardly the enlightenment you make it out to be."
I really can't understand how anyone who does sysadmin work and any amount of programming can say this.
My mastery of computers grew organically. I started out with "power user" stuff. Namely, tweaking my computing environment (Windows, at the time) to my liking. As I tweaked, I inevitably broke things, which meant I had to learn to fix them. It wasn't long before I realized that being able to fix computers when something goes wrong was a valuable skill, so I developed that skill further.
I had a friend who wrote simple programs in VisualBasic from around the time that Windows 3.1 came out. He wrote a baseball card database application that we both used to catalog our collections. From that, I understood the utility of programming as a means to solve problems that were imporant to me, but I was too consumed with other things. It wasn't until the internet gained popularity that I started to write programs. What I'm getting at is that I was late to the game. I was in to computers for a long time before I had much interest in programming.
My skills as a sysadmin were pretty decent to begin with, but I became a troubleshooting superman as my programming skills developed. It literally changed my entire perspective, and even my methodology, when troubleshooting. I could read a stack trace! I could load up dump files, import symbols, and dig around to find the offending file, which would lead me to a specific application or a device.
Even at a very basic level, learning to program taught me a tremendous amount about how software works. Those insights led me down the correct paths when working as a sysadmin. I began to understand not just what to do, but why.