There are only 3 shotguns that deal primarily Puncture damage; the Hek, Vaykor Hek, and Corinth. All other shotguns, including the Sweeper for Sentinels, deal less than 20% of their base damage as Puncture.
The Phage deals no physical damage whatsoever, so it only benefits from the increase in fire rate.
Prior to Update 12.4, at base level the mod was increasing Puncture damage by 22% rather than 15% and adding the Puncture damage based on the weapon's total base damage, causing unusually large boosts in Puncture damage as a result. These bugs have since been fixed.
The mod picture is misleading, as it implies that it adds punch through in order to puncture the Shield Lancer's shield, when in reality it increases the puncture damage type. while also, the shotgun featured in the image is a Sobek. which deals mostly Impact damage.