Actually moving the saddle like that would be silly, but people do move their seated position on the saddle when they change hand position. If you just lean forward to get more aggressive, the body angles scrunch up: pedaling gets harder, breathing gets harder. If you rotate your entire body forward about the bottom bracket, the angles are preserved and it's easier to drop the hammer while aero. Hence the phrase "on the rivet."

And bikes designed to be always ridden in aggressive postures often have saddle positions that are farther forward. Triathletes happily ignore the UCI rule about saddle noses being at least 5cm behind the bottom bracket, and for UCI-sanctioned TT events, some manufacturers sell saddles with short noses to cheat the rule.