Why do the members of the community support this method?
Historically, many accused were forced to face their trials in front of the entire community. In regards to the kingdom detailed in Stockton's "The Lady or the Tiger?," the punishment for the youth was no different.
One could assume that the semi-barbaric king found that his method of trial was best suited for his own kingdom. Instead of he being the one to lay down judgement, the king left judgement to fate.
In regards to the community's support of this type of trial, one can only assume if they supported it. Historically, arenas were used to both entertain and seek justice. Therefore, the community would come out to witness both.
Based upon historical footage and media's depictions of arena use, one could justifiably believe that the community did, in fact, support the methods employed.
One way to justify this assumption would be to examine the king. He is described as being semi-barbaric. One could only assume that his people were as well and that they enjoyed the possibility of blood shed.