Without knowing anything about the specifics of the conjectures that are falling, I wonder if this affair highlights an issue with the quality of the "open conjectures" that are out there.
There are a lot of reasons you might make a conjecture. Perhaps you are motivated by a search for structure: "We suspect that these objects behave like this, therefore we naturally imagine that...". But you might make a conjecture out of empirical evidence: "We checked 1 million examples, time to make a conjecture." While empiricism has its place in mathematics, it might not be the best motivation for a conjecture. As Bishop said, "Do not ask whether a statement is true until you know what it means." If you pose a conjecture based on empirical evidence, perhaps you should be thinking more about why you would imagine your statement be true in the first place.
I know some will think that this rings of goalpost moving, but when I hear about all these counterexamples spelling the end of mathematics as we know it, I just wonder how strong those conjectures really were in the first place. Again, I haven't actually dug into the specifics of any of the results so admittedly I could be totally off base, but I do know that not all conjectures are created equal.
Mostly the significance of a conjecture is that we don't know the answer. A conjecture is successfully resolved if it proven or disproven. (In fact, of the three big conjectures to be resolved in the past couple of weeks, one was proven and two were disproven.) The Jacobian conjecture, which is the one that was resolved this week, was largely thought to be false. It was only the fact that no one could disprove that raised the tantalizing possibility that it was true.
The reason the author is upset is not that the conjectures were false, but that an AI settled them. That's why so much of his post is about the loss of the human element.