In the Fourth Edition, monster do indeed have obvious abilities. Kobolds are shifty and hard to pin down. Giant crocodiles grab you in their jaws. Ghosts pass through walls and take half damage from attacks. Rats swarm you. Hippogriffs fly. Not sure why you would think otherwise.
Which 4E Giants can hit you and knock you back and which cannot? Some Hill Giants can do it, some cannot. Hill Giant Grunts can do it for pete's sake, but many other hill giants cannot. Hill Giant Grunts can do it, but many other bigger and more powerful giants like Frost and Storm Giants cannot.
I don't see the rhyme or reason for this. I think any giant (i.e. Hill Giant or taller, not just Ogres, specialized Ogres might have these abilities) using a really large weapon should knock foes all over the place. JMO. I think the same should be true for certain types of massively powerful Demons, and Devils, and Dragons. On the other hand, there should be some more subtle Demons and Devils and slim snake-like Dragons that don't use massive weapons that don't have this ability. But, Giants are massive by definition. They should almost all do this type of thing. I want Giants that are like the Cave Troll in the LotR movie. Not brutes that just swing for damage. The overhead shot knocks a PC prone. The normal shot knocks a PC back. A heavy duty shot knocks a PC back and prone.
I think the 5E game would be better served if the monster races were created first such that all monsters of each particular race have specific racial abilities. Then, create specific monsters of each race that enhance those abilities and add in new abilities.
4E's method of "maybe the designer remembered that Hobgoblins have this ability, maybe the designer forgot" is lacking in flavor and consistency.
On the other hand specific spell-like abilities are as wondrous and mysterious as those of any class's spells. BUT if the monsters' attacks are just the same as those from the characters' spell lists, then the mystery and surprise is lessened (and the logistics increased).
I don't think that monster attacks should always be the same as PC spells. I just think that sometimes they should be. I think the game is not served well if most abilities of most monsters are mostly specific to this monster. I think some abilities should be obvious and some should be recognizable (and even unrecognizable) spells.
I think that there is plenty of room in the game space for monsters that cast spells. Not just liches, but pixes and dragons and demons and devils and most humanoid monsters who have their own deities and their own prayers/spells and because they are humanoid, they have the same access to Arcane, Primal, Divine, and Psionic abilities.
I don't see why the physics of the campaign world should be segregated between PCs and NPCs. I do see a reason to make NPCs simpler and easier to run, but that doesn't require that NPCs cannot cast spells.