My experience is that it doesn’t even mean you played better than your opponent overall, just that you “made the second to last mistake”.
I also think there’s a larger element to reading your opponent in chess than many people give it credit for — what are they planning? …can you “tilt” them to make them play worse? Etc.
It’s only at the very highest level you should expect people to play nearly perfect games… and even those players discuss the psychology (eg, Hikaru Nakamura is top 50ish and discusses the role of psychology in high level chess).
That's certainly a thought from a brain. I'll hazard that parent's chess knowledge extends to "how the pieces move", possibly Scholar's Mate and its basic refutation (kidding, parent just read about Scholar's Mate on Wikipedia).
The game itself doesn't have intrinsic chance, but as with everything in life, the players do.
I do agree with your last point though, winning doesn't mean you played well, it just means you played better than your opponent.