Which means it can be done as either part of an action or bonus action.
Wrong. In 5e something is ether an action or a bonus action. You cannot use an action to carry out a bonus action, you cannot use a bonus action to carry out an action. Only special class abilities allow a character to do something using either.
From 5e Sage Advice:
Actions and bonus actions aren't interchangeable. In the example, the bard could use Bardic Inspiration or healing word on a turn, but not both.
Shove and Grapple aren't actions or bonus actions in and of themselves, they are things you can do in place of an attack. There is no Shove action. There is an Attack action and you can replace attacks given to you by the Attack action with shoves on a one-for-one basis. So anybody with Extra Attack could shove twice if they want. Heck, a Monk with Flurry of Blows at 5th level can shove 4 times. A Frenzied Berserker in frenzy can shove twice at 3rd level (bonus attack) and 3 times at 5th.
Switching shove from attack-option to a full Bonus Action sort of limits its use by the martials. It is also limited by the fact that you can't chose to knock someone prone and have them sit there. Shove prone+Grapple (they are now prone at 0 speed and thus unable to stand) is a great way on the TTRPG which you can't pull off here. Nor can you go on a shoving spree quite the way you can in the tabletop. Making the shove in general involve much further distances addresses that loss however. Overall, the impact of shove is different but not particularly any more powerful.
