That's... one interpretation of the Arachne myth. The one that, conveniently, is most favorable to the gods and, less conveniently, is the most incoherent.
Now, the coherent version... that's a bit less favorable. That one tells of a woman who was a great talent in weaving, so talented she caught the envious eye of Athena, who desperately tried to lay claim to her skills as if they were her doing, rather than Arachne's hard work. Eager to prove herself greater than a mere mortal, she challenged Arachne to a weaving competition, one that was horrendously unfair to begin with, given that Athena herself was the only judge. And, you know, a god, capable of supplementing the weaving skill of an immortal, with countless aeons to practice, with divine magic.
And yet, despite all these advantages... when the competition was over, even Athena could not deny that she had been bested utterly. And even better, she had done so with tapestries depicting the way the gods have abused and mistreated the mortals they lord over, throwing their follies into their faces in a grand and undeniable way. Arachne was simply better than her in this one, tiny aspect. And even Athena, wisest of the gods, was not immune to their characteristic petty wrath. She killed Arachne where she stood, the moment the competition was over. Not even smiting her with divine power, she simply took the weaving tool she had at hand and beating her to death with it, and tore her tapestries to shreds with her bare hands.
Then, moments later, she realized what she had done. In a moment of spiteful and envious fury, she had murdered a woman, an incredible artist, simply because she had fairly won a competition. Athena was horrified by her monstrous actions, and tried to atone for them by resurrecting Arachne as the first spider, becoming the progenitor of a race that would inherit her masterful weaving and spread it across the lands.
People claim that the story of Arachne is a cautionary tale about hubris. But it was envy that was the vice, here. It was envy that drove Athena to try and claim responsibility for Arachne's skills, it was envy that drove her to engage in a petty and foolish competition with a mortal, it was envy that drove her to a murderous rage. it was only in letting go of that envy, and ensuring that Arachne's superior tapestries would be spread across the lands for all to see, that Athena was able to even begin to atone for her monstrous actions.
But more than that, it's a tale of the irrationality of divine punishment as a whole. In the tale of Arachne, Ovid (the earliest known author of the story) takes a conventional tale of a mortal offending the gods, and turns it into a story of the gods’ arrogance, unjustified rage, and lack of mercy. Just as Arachne's tapestries were the antithesis of Athena's, so too is Ovid's tale an antithesis to the classical tale of gods punishing mortals. Ultimately, it tells us that the gods are not worthy of the respect they demand; they are petty and childish, their punishments are unjust and monstrous, and their greatest works can and will be surpassed by mortal hands.
In that sense, it's not dissimilar to some of the themes depicted in this story.
But this is why partial literacy in these ancient myths is so misleading. You look at Ovid's life, the context behind his stories, and you start to get a picture of a tale of envy, of the insanity of divine punishment, of censorship and lashing out at art that says things you aren't comfortable acknowledging. But you strip all that away, and simplify it for mass consumption, and you're left with a simple and banal tale about hubris and divine punishment, the polar opposite of the original. Like cutting a little square out of a tapestry, all you're left with is color, not a piece of art.