We practice from the perspective of the Buddha, not the Self.
At the start of the Heart Sutra, there is an exchange between Śāriputra, one of the buddha’s historical disciples, renowned for his wisdom, and Avalokiteśvara, the Bodhisattva of Compassion. Significantly, it is Avalokiteśvara rather than Śāriputra who, whilst sitting in zazen, realises that the five Skandhas are empty, and hence all suffering is relieved. You could say all suffering is relieved because Avalokitesvara, the five Skandhas and Emptiness are all synonymous.
Were Śāriputra, from the position of the self, to perceive the emptiness of the five Skandhas, suffering would not be relieved. The whole world would become suffering.
So, the suggestion is not that in zazen we see Emptiness, but rather that the five Skandhas see the Emptiness of the five Skandhas. And suffering falls away.