I was just wondering, whenever I visit any site it says pump the brake pedal three times then keep pressing it, what is the actual benefit of pumping it? And why not just press down the brake pedal and then open the bleeder screw without pumping?
You need to have positive pressure in the brake lines to get rid of any air; the pumping ensures you've built up enough pressure. And if the other sites you've read didn't mention it, you repeat the process (pump, hold, loosen bleeder valve while maintaining pressure on the pedal, close bleeder valve) until a solid stream of brake fluid comes out; that is the sign all the air is gone. And you can definitely tell the difference by the sound of the fluid coming out.
In past experiences I've found that intially pumping the pedal at the start was all I've ever needed and from there on only 1 push of the pedal is needed. The reason why is because when you repeatedly pump the brakes your taking that single air bubble and causing it to break up into many bubbles that must travel to the bleeder. Ultimately when it's all said and done I still prefer gravity bleeding starting at the left rear then right, then right front then the left. Takes a little longer but the advantages are you won't need someone to pump/hold the pedal and no chance of air accidentally being sucked back by late closing of bleeder when pedal is released.