No, I do not think that the play should have ended with Hamlet's death.
First, as Hamlet is dying, he tells Horatio to not commit suicide. Hamlet wants Horatio to stay alive, so that he can tell everybody the truth of what has transpired. It's Hamlet's dying wish that Horatio explain the events. It would be disappointing to an audience to not know if Horatio carries out Hamlet's last wish. There would always be a "loose end" hanging over the ending to the play. Horatio does indeed tell the events to Fortinbras thus fulfilling Hamlet's last wish.
The other part of Hamlet's last words to Horatio is that Fortinbras be made king. Without that last 50 or so lines, the audience wouldn't know if Fortinbras is indeed planning on taking over the throne. Those two reasons, Horatio telling the story and Fortinbras agreeing to the throne, are reason enough for Shakespeare to include those lines of text for the audience.
The final 9 lines of the play are necessary, because they show the audience that Fortinbras is treating Hamlet with a great deal of respect. He wants his body carried out by four captains and given all of the dignity of a soldier's death.