Not really. We use the 5e rules and mechanics, with some additional mechanics to modify gameplay. Improv is a large part of the point of playing TTRPGs, IMO, but that doesn't mean I'm just ignoring the rules and playing calvinball.
On the other hand, the rules do not matter or have any actual authority. Full stop. They exist to facilitate play, and that is the context they are used in. We use them because it's easier to do so than to just make everything up, and because we have found that different stories emerge when we freeform roleplay vs when we have constraints and oddball idiosyncrasies to bump up against and interact with.
There is a famous scene in the actual play podcast Adventure Zone, in the first campaign, Adventure Zone: Balance, that just would not ever happen in a story that doesn't feature the wierd idiosyncrasy that is the Magic Jar spell, or the way that dnd planes work, or some other stuff in that scene. Likewise, parts of that scene wouldn't happen in a game that makes a GM feel like improvising the fighter being robbed of their body and then possessing a wooden mannequin is going outside the scope of the game.
Again, while system matters, I believe that finding a mechanical framework you enjoy most, and then expanding that framework to encompass different genres, tones, themes, stories, etc,is worthwhile, because that is the primary sense in which system matters, IMO. The rules of a TTRPG are an illusion created by clever system designers to help a group facilitate satisfying experiences in the world of make believe. That's all they are.
Because of all of the above, I'm running a full campaign, or a game that doesn't stray waaaaaay outside of the wheelhouse of dnd (and so, doesn't feature any action of adventure or elements of the fantastical, pretty much) I will more often opt to adopt optional and unofficial and third party mechanics into 5e dnd, because my group enjoys and groks the basic systems of 5e DnD, and I can balance things at a glance when running it, and they know that when I say, "we are using XYZ optional rule" or "I'd like to try out using a success ladder of set DCs instead of coming up with DCs on the fly, and see how it feels" or "I've ported some concepts from various other games where players control more of the narrative than just their character's actions to a framework that I think will fit seemlessly into our play experience, are ya'll down to try it out?", they know what I'm saying, what to expect, what is being modified.
You keep saying that 5e doesn't have this or that rule, but what you're ignoring is that there is no one 5e DnD. My 5e game does have those rules, as does my fellow frequent DM, and as does my wife's when she runs the occasional game for her book club friends. In each case, the players do know what they're dealing with, and what the rules are, because they know our 5e. They know that in our 5e most important tasks will have multiple rolls, that they'll know how many before they roll the first one, and that they will have broad latitude in deciding what skills and how to use them. They know that languages are valuable both because lack of a language can cause disadvantage or simply cause a roll to be called for when it wouldn't be if they spoke the same language fluently, and that they can leverage a language just like they can leverage tools (xanathar's) to get advantage.
You can say all you want that "players don't know what the DC is so there is no rule except ask the DM" but I've never seen a single table actually run that way. I've never seen players uncertain of what their chances were or what would be asked of them or how a given task would generally be handled.
5e can be considered as much a toolkit to cooperatively make a game as it can be considered a game. That is what makes it better than some games for "hacking", that's why you can use the damage by spell level chart to balance damn near anything in the game with an understanding that the system only even tries to be balanced within a range of a couple spell levels in any given case, and it's why it has so many optional rules.
If I were to try and run Monster of The Week for an extended period, I would end up hacking it pretty inevitably, because yes, the system should not constrain the group. DM or players. IMO, of course, since apperently such caveats are required in this thread to not have one's words blatantly twisted.
As to the bolded text, well, that's the whole damn point, isn't it.
And you're wrong. Full stop. I didn't play this way in 4e because 4e actively fights against being played this way. You're essentially saying in one post that system matters, and in another that it doesn't. Which is fine. I'm doing the same thing, but inverted, but at least recognize what you're doing.
So, to sum up;
- System frameworks create differing levels of freedom, differing levels of prescription of process, and thus work better or worse for different groups in general. A group that prefers one isn't likely to get great results from the other.
- Mechanics can impact things like tone and genre, but because we are telling stories, prescribed bespoke genre mechanics are a bonus not a requirement for nearly any genre or theme.
- A game that presents a toolkit from which to build your game is not a game without rules, it's just a game where every table won't play exactly the same way.
- What is easy for one person is not as easy for another, and may be too hard to bother with for another, even if all three people are equally competent and intelligent.
- Pretty much no person's experience of anything is universal, so it is not reasonable to tell someone they can't do something just because you couldn't. It is reasonable to warn them of particular difficulties and pitfalls.