From my experience many humans have difficulties when it comes to learn from mistakes others made. Look at history, people make the same mistakes somebody did before them all the time, it's basically the main reason for history.
In the case of just killing a traitor, most observers/bystanders will probably laugh at the stupidity of the dead man and then do the same thing thinking that they are way to clever that this could happen to them.
That's why educational show trials have been invented.
Not to solve the problem of the misguided person who thought treason might be a good idea, but to solve the problem of others who might be inclined to do think that a change of leadership might be a good idea by preventing them getting the idea that a change of leadership might be a good idea in the first place.
A show of trail does no good when it doesnt fix the problem. Better to take it up from its roots
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