I can promise you that you wont make a single headshot with a trackball
I was getting more headshots on Call of Duty and Unreal Tournament with the trackball (with most of the guns except shotguns on CoD and the sniper + ripper from UT). I got a "Monster Kill" on UT that was entirely sniper headshots. I also snipe pretty well on Team Fortress 2. Once you get used to a
finger-operated trackball you will be very accurate. I use my index, middle, and ring fingers to control the ball. The sensitivity often needs to be adjusted, my rule of thumb is that one full left-right motion should make you do a 180 degree spin. I have good enough control on Call of Duty that I can use it to snipe with any gun, including the light machine gun with a thermal scope and variable zoom.
You CAN play FPSs with a trackball, however it isn't particularly easy. A regular mouse is far better for the purpose of games. I think it is a lot harder to stop spinning and aim with a trackball due to the inertia of the spinning ball, where as on a mouse, there is virtually no inertia, all you have to do is stop moving your hand.
I personally do better with a trackball. I use a Marble Mouse and the inertia is not a problem. The ball does not keep spinning unless you are trying to spin it as fast as possible while your fingers aren't touching it. I think maybe you haven't tried one because the inertia part doesn't seem to be accurate. Most trackballs have too much friction to stay in motion unless you spin them with great force. Like a lot of things, just like keyboards, it ultimately boils down to personal preference and what you are used to. Some products are better than others but everyone has different tastes. A trackball is the perfect fit for me and I wouldn't even go back to a Razer or a Steelseries, because I simply like the way it works. I can't guarantee that a trackball will make
you a better player, but since you can get a Logitech Marble or a Kensington Orbit for about $25-30 dollars, it's not too expensive to try. I will probably give away my old Marble or Orbit to someone will get a lot of use out of it.