Why does Simon's death prompt everything veering out of control? Surely Simon had very little influence on the boys?

It's easy to see how things get much, much worse after Simon's death, and I'm not going to spend time describing Piggy's death or the fire which follows in the final chapter. I think, though, you can see the seeds of what happens in the chapter which immediately follows Simon's death:

At last Ralph stopped. He was shivering.
“Piggy.”
“Uh?”
“That was Simon.”
“You said that before.”
“Piggy.”
“Uh?”
“That was murder.”
“You stop it!” said Piggy, shrilly. “What good’re you doing talking like that?”

There is a desperate need to come to terms with what has happened. THere has to be a way to make sense of the new situation: the goalposts have been altered. Ralph is now aware that they have murdered someone. The conch, in the face of such crimes, seems powerlessly weak.

“It was an accident,” said Piggy suddenly, “that’s what it was. An accident.”
His voice shrilled again.

You can hear the fear in what Piggy says. And Ralph admits it:

“I’m frightened. Of us. I want to go home. Oh God, I want to go home.”

The "darkness of man's heart" has become visible. It's the deed, not Simon himself, that demands a reaction. Yet, unlike Ralph, Jack reacts with strength, hunting, and solid leadership:

“But what happens if we meet—”
The chief waved his objection aside.
“We’ll keep along by the sands. Then if he comes we’ll do our, our dance again.”

Ralph and Piggy try to go back in time. Jack goes forward: and it is the harnessing - the acceptance - of that violence which leads to more violence.