Most of the NYC subway signaling technology is relay-based from the 1950s, so it's not quite a century old. Those lever machines at West 4th St shown by Business Insider are rare; almost everything is electrical.[2] The 1950s technology from General Railway Signal is quite good; it's just high-maintenance. Here's a 1950s era panel, running automatically and unattended during off peak hours.[3]
That technology really is quite good. All vital relays are open in the stop condition. All broken wires result in red signals. This is a basic design principle of classic railroad signaling. The mechanism for stopping trains[2] is brutally simple; a steel train stop comes up at red signals which, if a train tries to pass it, will be hit by a valve lever on each subway car which will directly trip the air brakes. The position of those levers is monitored by circuits which insure that there are always train stops in the up position between trains. If a train stop lever won't go up (ice, mechanical breakage, somebody or something holding it down) the previous train stop won't go down, maintaining protection, and signals will be forced red until the problem is fixed.
One reason it hasn't been replaced is that the newer technologies don't have as good a track record. The old stuff is very rugged, and is known to survive dirt, ice, snow, vandalism, traction power, vibration, rocks, water, and idiots.
[1] http://gothamist.com/2015/07/28/subway_steampunk_video.php [2] http://nycsubway.org/wiki/Subway_Signals:_Train_Stops [3] https://www.youtube.com/watch?v=RStJ621auaU