We disagree.
When a character rolls to hit and gets a 2 and misses then rolls a 19 and gets a miss, do you accuse him of metagaming if he draws different conclusions from those attacks? If he decides after the 19 missed "better get advantage or switch things up" but didnt after the 2 missed, do you bring down the metagaming thunder?
I'm a pretty hard core simulationist, and I believe that the purpose of game rules is to create a model of the world. For most things that we are modelling there should be a certain verisimilitude to the real world. That is the mechanics should model our own life experiences so that the results are familiar to us. You implicitly understand this when you write:
In almost every task i have attempted and most challenging tests or tasks, i have left it with a good ferling, a goid sense of how i did.
So I think we are in pretty strong agreement about what the process of play and the rules (ei, the fortune test and the resolution mechanics) are trying together to achieve.
Now consider the case you offered up as your first example, that of employing a weapon. When you employ a weapon against a target, isn't it the case that you get very immediate feedback regarding whether you hit or did not hit the target? You can in fact see that your blow landed, or that the arrow hit the bull's eye or otherwise know that you did well. So it's not at all wrong and indeed feels correct that a PC should see the result of his dice roll and have some feel for how he's doing.
But my contention is that not every real world test works that way - which is in fact something you seem to concede.
On occasion i was wrong.
Read that as "i misread the DC".
That's one example and one possible explanation but I don't think that is the sole explanation. Quite often in life you get things wrong, and sometimes you have a great feeling that you did well on a task which has a low DC and yet it turns out that you did poorly. There is actually an interesting psychological phenomenon called the Dunning-Kruger effect where the less skilled you are at something and the less well you do at something, the more confident you are that you did well at it.
With a little reflection I don't think it will be hard to remember cases where you thought you were doing well, only to discover you didn't. In my case I think of things like math exams where I made careless mistakes, programs I wrote that had stupid thoughtless errors in them, and playing 'pin the tail on the donkey' and thinking I was doing well when in fact I was wildly off. I don't think you can explain those just as 'misjudging the DC'.
It's my contention that in cases where the character can receive no immediate feedback as to how well they are doing, that the player should also receive no immediate feedback as to how well they are doing. So for a 'move silently' check or an attack roll or a climb check, sure I have no problem with the player rolling their own dice because as in real life, that character should be getting some sort of immediate feedback.
But in real life you have no way of knowing how well you are searching something. There is plenty of psychological experiments showing that people wildly misjudge how well they remember a scene or took in the details of something. How could you possibly know you were missing some obvious detail? I mean, haven't you searched a refrigerator before and felt you'd done so thoroughly, only to discover upon second look that the thing you were looking for was at eye level on the top shelf right in the front? Someone else pointed it out to you and you were like, "Oh that was obvious." It wasn't that the DC was higher than you thought it was, you just rolled poorly on your search check and missed an obvious detail. That sort of thing happens all the time. I would think life itself would refute your assertion without me having to.
A character swings his axe and rolls a 2. Do Many GMs narrate that as "for some totally unknowable reason your swing misses and you have to act in character like it hit or we are gonna have a metagaming problem"?
Do you now have some idea how ridiculous this example is? Do you think I narrate an axe swing like that or ever roll the dice for a PC's ax swing in secret away from the player? To be quite blunt, you think you did really well in offering up this as an argument, but you actually "rolled a 1" and completely fumbled it. Yes, obviously for an axe swing a PC gets immediate visual feedback on the axe swing so there is no harm in giving the player the same immediate feedback. But that example only serves to show just how little you've actually considered this question.
PS: "i have left it with a good ferling, a goid sense of how i did" - Did you have a good sense of how well you'd spelled your response as well?
