And the president can help to influence policy. For 100,000 online signatures, the White House gives an official response. It also tells the government what people everywhere want That seems a lot more effective than calling your congressman in Montana.
Maybe you'll recall this petition: https://petitions.whitehouse.gov/petition/make-unlocking-cel... and how a bill was announced soon after. Sure, the petition didn't cause the bill to come about, but it helped to show the will of the people and open up government a bit. Is that ineffective?
The government is far more complex than the Constitution would have you believe.