In 5e, Shove is exactly an Action...it replaces an attack. Which means it can be done as either part of an action or bonus action. A 5th+ Martial could do two of them in a single Attack Action. Having it as a bonus action here does not bother me at all and it is one of the most fun things in the game to be able to shove someone off a cliff.
I tend to warn against being too precise in details when adapting a thing from one medium to another. Something that's fine on the tabletop doesn't work as well in a computer game. There are some things that need to be simplified/streamlined and some that will be made more complex (especially the sort of optional rules that would slow down game play on a tabletop but which the computer can do in the background easier.)
