Well, the first thing is to think about symbolism and imagination, and behaviour. What we want most, as DMs, is for players to feel something and to do something in response. Say I simply tell a player that they can make a Dexterity (Thieves' tools) check and if they fail, they will take 8d6 of fire damage. That alone can be enough to shape their behaviour. And this is what RPG is about. It's not about neatly parceling up ability checks with immediate consequences, or lavish narration. It is about a player thinking "Golly, I have 18 HP... if that thing goes off I am toast!" and changing what they want to do. The things you describe are representations that stand for more. I can tell my players - "It'll take an athletics check to climb that wall" - and that really can be enough for them. It's an imaginary wall. Imaginary walls have no qualities. What they need to know is that a fall from near the top is 12d6 of bludgeoning damage. The word "wall" is not a wall, and most certainly the words "sheer wall" are not a sheer wall. We are working with symbols, and imagination, and we aim to make players feel something and respond.
Except for the bit about "lavish narration", which is
not required in games we run, this is very much in line with how I run ability check situations, when they are called for. Just add a DC in front of your phrasing such as : "make a DC 20 Dexterity (Thieves' tools) check and if [you] fail, [you] will take 8d6 of fire damage". Now they know the difficulty and stakes to make that informed decision. And the player then has the opportunity to come up with a different course of action because the assumption is that their PC is a very capable adventurer that is often able to discern the difficulty of the task before them.
With that in mind, I think I'm hitting both your wishes from the first paragraph to shape behavior and your second paragraph where mechanics add to the tension. This is happening as a result of the player being reasonably specific about what their PC is doing in the game world. It is the DM's responsibility to invoke mechanics when appropriate. It's the player's most basic role to "describe what they want to do."
At
my 5e table, players invoking "I use Arcana" might be using symbols, but they are not using imagination in the context of the scene to the best of their ability. We're all more likely to feel something and respond when a player describes what their PC is aiming to do and why, maybe invoking something from their character's background, maybe utilizing something from their equipment, or their personality, or race or class or whatever.
I just find "I use Arcana" to fall short. The goal here is not for the player to seek a mechanical solution necessarily, but to seek an imaginative one that engages with the game world and helps build immersion for all at the table. First person, third person, a few sentences, a short phrase, it doesn't matter as long as it is reasonably specific. "I use Arcana" is not specific enough for me, as DM, to adjudicate without making assumptions about what the PC is doing in the scene - so I'm going to ask for more detail.
The second thing is think about game as game. Players are psychologically capable of experiencing tension just from the mechanics. Roll a die and get 8+, or take 8d6 fire damage. That stands for something that can just of itself - as play - be tense. Just of itself, it can inform behaviour. Game dynamics can have in themselves a great deal of psychological effect. Perhaps you know the "thought-worm" mechanic that some horrors can use in Earthdawn? It is a beautiful mechanic that just through the way it operates, can lead players down a troubled path. Often the game mechanic is capable of doing the work: that's generally what they are there for! A very interesting question is - what psychological impact do differing probabilities of success have? What is the difference between making one roll - live or die - or a dozen interdependent rolls? How does overwhelming damage feel relative to incremental damage. To think of game mechanics as shorn of meaning unless slathered with verbiage is to terribly misapprehend game qua game.
Again, off the mark if you think our gameplay must be "slathered with verbiage". It just does
not represent game play at my table or, likely, any other table that employs this method of insisting that players engage with the game world rather than just the naming a skill.
And this is aside from some of the very misleading - or let us say, idiosyncratic - applications of the 5e mechanics in the PDF.
It jives with the rules, both new and old players pick it up very quickly, and it produces a fun result at the table so I'm really not sure how you can label it "misleading" or "idiosyncratic" -- other than perhaps it may seem somewhat foreign to your own experience with 5e. Hopefully my explanation under your first paragraph helps you realize this style is not as foreign as you have been trying to claim.