Pushups should be done with shoulder blades protracted and depressed to some degree and NOT shoulders neutral which is the general take. Protraction relies heavily on serratus anterior engagement. Pushups might have to be done many times before learning proper muscle activation. But working on it now makes PE 3 easier.
Pelvis should be tilted posteriorly which equals a true neutral spine.
Notice that the athelete is resting on the toes and not upper side of the foot like PE 3.
The 5 x 15 rep scale is focusing on building strength endurance which many people have a hard time gaining. There have been many takes on how to overcome plateauing. Strategies like greasing the groove (1-2 sets of pushups every day), adding rest time and eventually reducing that to mastery requirements, and breaking up a minicycle into two making progression requirements more favorable.
Pushups relies heavily on serratus anterior and triceps, but also involves the pecs, deltoids and traps. Core will also be moderately involved.
Pushups serves as introductory strength which should carry over to dips and handstand pushups and HBH itself. It also serves as a scapula mobilization drill where you properly learn how to to protract and depress and then building up basic strength in that range of motion which. In the end pushups should allow you to safely and effectively progress you well into dips.