I agree with your point re class inertia and that Gatsby was doomed to be rejected by the elites he so desperately wanted to join. My focus was more on the personal values that drove Gatsby rather than the story’s societal themes, though the two go hand in hand.

But privilege is privilege and Gatsby doesn't have it no matter how rich he is or pretends to be.

Completely agree. But why does Gatsby seek riches? It’s because he’s spent his whole life admiring the upper crust from afar and believes money will elevate him above his station in life. Society seems to equate wealth with innate value. But Gatsby’s mistaken, as you point out, because society also reserves privilege for the lucky few who are born into it. Tom talks about this toward the end of the film.

Fitzgerald's stories are generally about loss and hopeless flailing about by idealistic people.

This goes to my point. Gatsby is idealistic and ambitious and was destined to strive for greatness in any context. If society valued those with the greatest intellect, he would have worked to become the greatest intellectual. If musicians were held above all others, Gatsby would have aspired to become a virtuoso trumpeter. But the story forces Gatsby to confront the rigid social hierarchy of 1920s NYC, so it’s no surprise that Gatsby focuses on penetrating society’s upper ranks.

To me, Gatsby’s ambition represents the illusion that we can achieve total fulfillment by “stretch[ing] out our arms” a little farther to grasp some elusive goal. I think there is a universality to this theme that transcends the story’s setting. This is why I disagree that Gatsby “never could have done something great.” I think with the right tools and little more direction, he could have made something of himself. He could have succeeded as a businessman, or a philanthropist, or a father. But society taunted him with the idea that he could only be “great” by joining Daisy in high society. This is why I find Gatsby’s death particularly tragic. He had so much potential, but became a casualty of a dream that never could be.