Chain-free bike designs (and other major hardware changes) come up periodically. Typically, they're belt-driven. While chains and cables for brakes seem inferior to cables and hydraulic brakes, they degrade for more gracefully. It's not a perfect solution, but it's a reasonable local maxima for most people. Professional racers can get away with equipment that's wildly impractical for commuters, mountain bikers, etc., though.

"Although, that would presumably be the least of your worries if a fast-moving, taught, metal wire lashed off its piston next to your leg. Ouch." I've had a couple chains break or skip off (especially on fixed-gear bikes, which don't have a derailleur or chain tensioner to regulate chain slack), and having a taut wire whip at my ankle sounds like a terrible idea. Getting a bit of grease on my ankle or pant leg is rare, and at most a minor annoyance.