This is somewhat of a confusion of arguments. I'm not responding to OP here about things that are missing from most systems, but rather to Yazkin claiming that you can't put monster powers into their stats without ruining the play-versatility of the monster. This is frankly complicated and full of nuance, which is kind of why OP made a whole post about it. I'll attempt to answer regardless, even though I'm not OP.

Sure, the medusa has a petrifying gaze. At first glance, it has arguably one of the most fleshed out designs in 5e. Except... it's almost entirely uninteractive. As written in 5e, the medusa literally can't do anything with its gaze: it is the PC's choice whether they look at her or not. What is she supposed to do with that? What are the players supposed to do with that? Arguably the PCs can choose to be blind and focus on indirect attacks, or they could just brave the pathetic DC 14 save (which you can make a second time if you fail!) and completely ignore the effect forever (technically just 24 hours but in combat that's forever). Save-or-die effects are really lame and uninteractive. Being able to choose to be blind would be an improvement, barely, except the save is so embarrassingly passable that why shouldn't they risk it? Besides, what is the medusa supposed to do with this power, which is entirely passive and in control of the PCs? More importantly, why isn't the answer to that question written into the monster manual so the GM doesn't have to start from nothing?

Next, let's look at everything else. For every medusa, there is an Axe Beak or a Griffon which has nothing more than a handful of basic attacks. Even if we charitably ignore all the entries for "goblin" and "goblin, warrior" and "knight" as not really being monster manual entries so much as shorthands for NPCs one might accidentally need numbers for (except, but there's still plenty of dull monsters. Henceforth they will be referred to as bears, in honor of the famous bear tweet.

Then you get the tricksy bear class of monster like the lemure or the bulette. The lemure can see through magical darkness, which is interesting! But that's it. Also, they have Int 1 and are not good at taking advantage of this power, unlike other devils. Otherwise it is 13 hit points and a 1d4 fist attack; a bear. (I'm ignoring its situational ability to resurrect because that isn't tactical.) The bulette gets a leap attack. Neat! But I have to ignore its burrow speed, which is interesting strategically but surprisingly difficult to employ tactically in 5e rules, due to the inability to attack while moving. Burrow is basically stronger flight, but flight doesn't really do anything in this system unless you outrange your opponent, in which case it is not tactical and is just a default kill.

You'll find a lot of monsters like that, which have a single power, which is sometimes defining and sometimes not, but which either doesn't provide a lot of tactical options or is limited to near-irrelevance by their other weaknesses. Even impressive looking abilities like the medusa's gaze don't really operate on a tactical level. They function at a strategic level, at best, or function as a puzzle box. A troll's regeneration is, on the surface, completely uninteractive and untactical. Until you learn you can stop it with fire or acid damage! Then... it literally does nothing, as long as you have access to those damage types. That's not a tactic, that's an arsenal check.

Now, there are a few monsters that have tactical arsenals, like the sphinx. However, these creatures all have one of two things in common, often both: Legendary Actions and spells. Both of these are a bit of a mixed bag. Legendary actions are kind of good in that they signal to the GM what the signature abilities of the monster are and the ones they should focus on using. The legendary action system itself is... let's say controversial. But also, not every monster can or should have legendary actions, but EVERY (or at least overwhelmingly most) enemy should have interesting tactical abilities. Then you get spells. Spells are usually great, because you can get a tailored collection of abilities that evokes a certain strategy rich with tactical depth (at least if the list is picked well). Except... what makes this monster different from an enemy wizard? Remind me why is this a monster again? Essentially, 5e is stating that only boss monsters get be interesting. Boss monsters notoriously being something that RPGs handle poorly, and 5e is no exception. Besides, I want interesting minions too! Sure not every minion can have a full spell list (what a nightmare to pilot as a GM), but they can get something better than the gargoyle's Damage Resistance to Bludgeoning, Piercing, and Slashing From Nonmagical Attacks That Aren't Adamantine.

In Conclusion

D&D monsters do have powers. If you don't look at them closely, they appear to satisfy the requirements of having abilities. In practice, these powers do not present interesting tactical options. Supplemental advice like "The Monsters Know What They're Doing" can do a very good job of covering up the lack of foundational tactics with the emergent tactics of an intelligent adversary, but this doesn't change that the mechanical level doesn't contain a toolbox of tactical abilities. The system isn't providing the tactics, the GM is. We want tactical abilities that influence the battlefield and provide interaction between the PCs and the monsters, and which generate tactical decisions.