We're not talking about remote control drones, either--though they're becoming more and more popular for the same reason. We're talking about robotic fighters. The Ministry of Defense in the UK recently unveiled the most advanced one of these, called a UCAV or the Uncrewed Combat Aerial Vehicle.
Right now, the plane is limited. It is able to run only missions that are preprogrammed into its system. And while it can spot targets, it can take them out only if it requests permission and permission is granted from a human (that all might sound pretty comforting if it weren't for the fact that drone aircraft can and have gone AWOL). The point is, it turns out more than five people should have gone to see the movie Stealth because it was about the inevitable future: The US Air Force put out a report last year outlining how they could replace every damned aircraft they own with flying robots in the next 40 years.
It's definitely a good idea to remove "conscience" from the mix when dealing with giant missiles.
Beyond the fact that modern militaries are less and less tolerant of losing troops because of the political backlash that comes with it, there are just physical limitations at play: A soft human body can only withstand about 5 g's (that is, five times the force of gravity) before losing consciousness. A trained pilot, with a special suit, can maybe get up to 9 or 10. Aircraft designers have to start from the premise that too many barrel rolls will leave the pilot bleeding from the eyes.
But take out that limitation and they can start drawing up a radical new plane whose performance is limited by the strength of titanium and carbon fiber, rather than the tiny blood vessels in a human brain. If nothing else it should make air shows of the future way more awesome.