Not all monsters that a DM might want to have push or grapple a PC will have such abilities. Like, sure, a wolf can just do it on a hit, but sometimes you don’t want a wolf, you want an orc, or a bugbear, or whatever.
I think it’ll vary quite a bit from table to table. Personally, I like to have monsters favor enabling themselves and/or their allies to attack with advantage over attacking without advantage, so I anticipate them making a lot of unarmed strikes to knock a target down and gang up on them. For DMs who prefer to just have every monster use its higest damage attack every round, there may be fewer unarmed strikes being made.