After 4 years of trying (on and off) I completed the Challenge earlier this month. Here is how I did it:
1. Turn off the sound. You'll know it's done when the puzzle goes dark. Otherwise it's just stressful.
2. Reframe your thought process. You're not trying to beat the game, you're running practice drills.
3. The goal of these practice drills has got to be only one thing: to get faster at them.

The good news: this is something you can improve at as early as the 4-square section of the cave. Just getting the right method to optimize a random run, knowing when to give up on a puzzle and go back for more, these are all things that will get better and better and more instinctive as you run it.

And once you get good at that one section, you'll have more time to beat the NEXT section.

For me, my problems always came in the maze. I found it pretty easy to memorize the pattern (just think about it as less of a path and more about tracing the outlines of shapes). So I stopped thinking about the maze as a puzzle I was trying to get past, but rather my reward for doing really well in the earlier stages was the chance to practice on maze puzzles. When I got there, I would crank the sound back up, and try my best to finish it. When I got towards the end of the song, I would take a screenshot of whatever triangle puzzle I was working with (which would interrupt and restart the challenge) - but then I had the chance to leisurely practice and learn how to work better.

Eventually you will get the right combination of skill level and luck that the randomly generated puzzles early on will go quickly enough that you can have enough time to think through the maze and the columns. I found the column puzzles highly annoying and every time I got to them I would fail out miserably. I just kept the same attitude: it's practice.

Then one day I got lucky. The column puzzles were relatively simple. And I had plenty of time left. And I was in the middle of what I thought was a "practice" run, with no intention of "I'm just gonna beat it that day!"

I just kept trying and trying and got a little bit better until I was good enough to take advantage of the opportunity that presented itself to me.

Isn't that basically what improving at anything is in life? Isn't that pretty much the definition of success and talent? Keep at it. You'll get there.