Powerful Build only say it work for carrying capacity and weight you can push, drag, or lift. Nothing more, nothing less. So it don't affect moving a grappled creature.
So RAW it don't work. But if as a DM you do the rule of cool you may consider it ok.
No: Shoving or Grappling require an Attack action, not the Multiattack action.
SevenSidedDie's answer clearly explains why a creature must take the Attack Action to make a Grapple or a Shove attack and can not substitute one of the attacks granted by the Multiattack Action.
Moreover, the Sage Advice Compendium confirms SevenSidedDie's explanation:
Is the ...