Why kill instead of question? When the crime coefficient of an individual rises high enough, the Dominators are set to kill them when possible, seemingly by exploding them from inside. This causes some problems:


The police lose their chance to question their suspect, possibly hampering ongoing investigations (including information on Makishima).
It is ethically questionable to destroy someone's body; Japanese culture would probably want an intact body for a proper mourning and funeral.


Of course, maybe the Lethal Eliminator just looks cool enough, but are there better explanations?
The police lose their chance to question their suspect, possibly hampering ongoing investigations (including information on Makishima).
  


The premise in the series is that Sybil can identify any criminal with a simple mental scan, and most of the city is scanned constantly for stress levels and such. Hence, killing off a criminal won't hamper any sort of investigation, because all criminals are found just in time. In other words, there is no such thing as organized crime, because the rigorous scanning measures always detect everything in the city.

... Which is of course flawed. Individuals like Makishima can't be detected. However, as we learn later in the series, such "criminals" have a special role in the system, so we can ignore them.


  
  It is ethically questionable to destroy someone's body; Japanese culture would probably want an intact body for a proper mourning and
  funeral.
  


Sybil has crafted a peaceful society by making objective, calculated decisions. It has determined that an individual with a criminal coefficient greater than 300 has no salvation, and is only a danger for everyone else. I think it is a key feature of the series to display a society that is culturally different from ours.