Puzzle help

I'm trying to understand if my reasoning in solving this puzzle was sound.

1. I eventually understood that the only way for me to win was for my king to advance to d5.
2. For that to happen I need to have a pawn on b5 to take away the c4-square from White's king.
3. Once I understood that, I understood that I needed my pawns to be in a position where it's White's turn and he effectively has to retreat his king.
4. So I lock the kingside with 1...hxg3 2.hxg3 gxf4 3.gxf4 (The puzzle ended with 2...gxf4, hence this thread)
5. Now I can push 3...b5
6. If 4.a5 then 4...a6 and White's king has to retreat to the third rank
7. If 4.axb5 then 4...cxb5 and still White's king is pushed back

Is there a nuance or insight that I missed?

Sounds all very reasonable to me. I might add that in point 7 after 4. axb5 cxb5 Black's a-pawn becomes backwards, so it might be good to calculate a few moves further. After 5. Kc3/e3 Kd5 6. Kc2/e2 Kd4 7.Kd2 the reserve tempo 7... a6 brings White in zugzwang again and decides the game.

I have not turned on an engine, but this seems to be a straightforward win.

Ok, I think your thinking process is ok but just to solve puzzle. You are not exctracting insights to solve these types of positions faster. Example ( and there are probably zillions of insights more but one of the first I've noticed) H-G and G-F capturing recapturing helps you to keep initiative, so if he has to recapture ( take optional in chess!) but if he has to, you have time to create a new threat or manifest your idea like in this puzzle. So, you can exctract "a little more benefits" If you will focus to notice patterns which may help you to understand and spot ideas faster next times.

Thank you for these helpful comments.

Your thinking process it is right, but it seems you dont know about opposition.

It looks like you kinda understood the opposition concept given the position, but not because you already know about opposition. You just got it on the fly.

Opposition is a global concept that has to be tattoed in your brain by default.

So the train of thought should be like.

Oh, its an opposition exercise. I must win opposition.

How do I gain it?

The exercise itself has some other subconcepts, like initiative, which you have to maintain too.

To solve the puzzle, yes, you need to advance to gain control of b5, and the king is screwed, but since white king has other moves he can make, you need to neutralize those first, so you need to clear the kingside first. hxg3 forces hxg3 else you promote. Then gxf4 is correct.

Then now, again, he has to capture or lose opposition. So gxf4. And now, a6 and he cant avoid losing the c4 square.

But learn opposition. At the end of the day you solved correctly, but if you manage to encapsulate chess into concepts that you can apply in specific situations rather than figure it out on the spot, you will save a lot of time.

Imagine you were playing 5+1 and had 10 seconds to figure that out. I would have spent 1 second to know is an opposition drill, maybe 5 seconds to figure out the correct sequence and premove my way for the win as the rest is also straight forward. Thats why you also need to learn the king + queen mate. You just go in auto pilot. Learn the concepts.