Today, Allan over at Arlington's Stihl dealership called to let me know that the parts arrived to fix our chainsaw. It turns out that the handle I broke on our MS-271 is part of the same housing as the fuel tank, which makes the repair pretty extensive. Rather than dive in blindly by myself, I opted to have them fix it under one condition: that I could watch and learn.
Normally, I have no problem tearing things apart on my own, but I've never worked on a chainsaw. I figured it'd be worth it to see at least one teardown by a professional. It was. Allan, and the other tech, Johnathan, were both super friendly, and answered every question I had during the repair.
All told, it took about an hour, and it required a significant amount of dismantling. As a result, I got a good look at how pretty much everything works on our saw. Now, I feel pretty confident I could fix this thing if I had to. Hopefully, I won't need too, though!