TL;DR - pages and pages of word salad and pointless math which boils down to – the gambler's fallacy is a fallacy if the coin is unbiased, but maybe it isn't.
Many things aren't independent. The article mentions the odds that it will rain on day depends on whether it rained the previous day. Likewise, the odds of a bus arriving in a city with a well-organized strict bus schedule is less if a bus just recently arrived.
How about blackjack? If I get an ace, are the odds still 1/52 that my next card will be an ace? Of course not.
How about the lottery? Are the odds that a scratch-off ticket is a winner different if the previous ticket on the roll was a winner? I have no idea, but I wouldn't be surprised if the lottery runners introduced such shenanigans.
I think that's the author's only point - that we shouldn't always assume independence.
But it falls apart a little because it seems to be talking about coins (even if it's actually talking about koins), which is a domain that starts from the premise that events are independent.
And that premise is empirically false as noted by the author.
Now the question is whether it's meaningful to accuse people of committing fallacies in hypothetical scenarios where perfectly fair coins are real.
The author is not making this point, and early on, he calls coin flips "Markovian" which is synonymous with "memoryless". Independence is assumed in this article.
Independence is not assumed in the "sticky" or "switchy" states in the article. The probability of the next flip depends on the previous flip.
You're right about the article.