Gamblers fallacy is me rolling a die 10 times and so expecting the 11th time to be anything other than a 1 in 6 chance.

That isn't what I'm saying. I'm saying if you roll a die 10 times, the chance of any one of those rolls being 6 is greater than 1 in 6. It is statistically likely if you roll a die 10 times, that you will roll a 6 at least once.

That's not the same thing as making progress though. You never get closer to rolling a 6.

I said "kind of", because there isn't a direct link no.

But my chances still increase. If I roll that die once, I know my chances of rolling a 6 are poor, if I roll it 10 times the odds would be better, a million times almost certain.

If my chances of success are steadily increasing, why is that not progress? (especially for an algorithm based solely on chance).

Your chances aren't steadily increasing. The chance you finish within a week is higher than the chance you finish within a day, but your chance is no higher after a day than before it.

Consider an alternative algorithm based solely on chance. Pick an element uniformly at random and if it's larger than the element after it, swap them. This algorithm has progress, as the array becomes monotonically more sorted over time, decreasing the expected amount of time until the algorithm is done. This is what it looks like to make progress. Your algorithm does not have that feature.

"The chance you finish within a week is higher than the chance you finish within a day, but your chance is no higher after a day than before it."

You seem to be ignoring the fact you've just been spending a day rolling die, where if you had have rolled a 6 you wouldn't be back the next day.

That's survivorship bias.

I never said you couldn't eventually win. We're talking about whether you make progress leading up to the win. You could win at any time, but you never get closer to winning. If you had a progress bar based on how long left until you win, it wouldn't move until the moment you won.