New Zealand beat Australia 27-24 in the final, with 18-year-old Rieko Ioane scoring a dramatic last-gasp winner.
Scotland were beaten 35-12 by Canada in the semi-finals of the Bowl competition for teams finishing third and fourth in their groups.
Wales were also beaten by Canada, losing their quarter-final Bowl match 32-21.
Fiji lost 14-12 to the eventual winners in their semi-final meeting but beat South Africa in the third-place play-off and are top of the series rankings on points difference.
The Fijians, South Africa and New Zealand are all tied on 69 points after four rounds of the 10 rounds of the championship.
However, New Zealand are the subject of an investigation after claims they had eight players on the pitch at once towards the end of their pool match against the hosts and could be docked points.
Australia were not behind at any point in the final until Ioane's late try.
"They had us on the ropes in the first half," said New Zealand skipper Tim Mikkelson.
"We know from last week what kind of lift you get from your home crowd. We just needed the ball, we're a difficult team to beat when we have the ball."
England were also beaten 19-14 by Argentina in their Plate semi-final but had injury problems.
"It hurts, and the boys want to do better than this in tournaments, but they fought and fought against the odds so I can't ask for any more," said England coach Simon Amor.
England suffered a 17-12 extra-time quarter-final defeat against hosts Australia in the Sevens World Series.