This was indeed a huge work, thanks for the help @RedMountainFox.
Kf3 is indeed assisting the pawn without closing the way to promotion. For quite some time the king will have to hide in e1 to win the game... Could you think alone of the Lucena position starting from the basic opposition concepts?
I appreciate indeed to see the way you think. When I check for example speedruns it is difficult to follow. It is usually masters, and even the less strong do everything naturally, and you do not grasp the nuances and how difficult could it be to evaluate different candidate moves... when they already see the best move at first glance.
I will try again the drills. I do it every now and then. 
I know relatively well (better than many other concepts, at least) the opposition principles, but reviewing it can only help.
Ah I see. The point of the positions I gave is that you have to understand opposition as well as having the understanding of how to promote the pawn without drawing. Once you understand those positions, you understand that it is important to not push your pawn too early, the importance of having the king to assist promotion, the importance of not allowing the enemy king in front of your pawn or enter your opposition, and the enemies best chances/ideas for getting a draw out of a completely losing position. Here is an additional one to supplement the other ones.
This is a known draw position if our king is behind the pawn and the enemy king is in front of the pawn, regardless of transposition to any rank, and regardless if it is the a or h file (I mention those files because there are exceptions in certain king and pawn endgames), and regardless of who moves first.
It might seem like those positions I gave are unrelated, but once you understand the basics of those above endgames, you can now consider what white would want if in your given position it were not black’s turn. Analyze their threats/ideas. I would find that white would play Kd1 to attempt to get in front of the pawn as that is their best chances at a draw (moving the king away provides no fight and no chances and is not logical). Engine provides Rf8 - If you respond to Rf8# with Ke3, Re8 and you can not push your pawn because the king is in the way and despite having checks against white’s king, you find that there is no progress to be made. If after Rf8# you move your king away, black will repeatedly threaten to capture your pawn via Re8 and force your king to keep contact (you can also defend with rook but white’s king is too close and there is no win there but requires rook endgame knowledge). As a generality rook endgames are often very drawish.
SO now you understand white’s plan. How do we prevent it and implement our plan? Do we have a plan? Yes we have a plan because we know we need the assistance of our king to promote so we play Kf3 and we can deduce not Ke3 because that gets in the way of the idea of promoting. We are also not afraid of Ra3# after Kf3 because we know that we can definitely promote because we have reached the 6th rank and our king is there to help and white's king is not in time. We are not afraid of Kf3 Rf8 because we just go Ke2 and white is out of checks for now. I can't go much further as I've put a lot of effort into typing this but I hope that makes it clearer of why those are important. The ideas transcend the static positions.
https://www.chess.com/practice/drills/endgame-fundamentals
I highly recommend working on the first two drills here in the practice/drills section. That is what I worked on a couple of weeks ago and now I can get it 100% every time because I see the patterns to avoid allowing the enemy king to enter my opposition, I know when I want to enter opposition, I know not to push my pawn too far too early, I understand adjacent opposition, and I understand what to do once I reach the 3rd and 6th rank. I guarantee that these drills are extremely challenging until you master the technique. You are completely winning but if you lack precision in an endgame, you get a draw. You must play perfectly.
Kf5 is an example of what I mean by adjacent opposition btw, it's how I remember it and I'm not certain if there's a name for it but that is what allows us to advance our pawn after re-entering opposition if the enemy defends with best play.