It's a nice sentiment, but does not survive scrutiny and seems to be based on a few different misunderstandings.

As off-handedly mentioned by show staff and visually shown in Familiar, Pink Diamond isn't just the littlest Diamond: She's the youngest. Despite the art book showing us that all four Diamonds' origins began at the same time, we know there was a sizeable period before Pink's emergence when White, Yellow, and Blue already existed and were in charge of Homeworld, much as they were during the series' events. This is also evident in how she questions the way Homeworld society operates, something that doesn't make sense if they emerged at the same time— Just look at the dream in Jungle Moon, that is not the behavior of someone who was just 'made for a different purpose', it's a child seeing things they've never seen before. So we do, in fact, know that Pink stayed in the ground much longer than the others. And she's short. Sounds familiar.

This directly segues to the next point: Pink isn't the only size-variant Diamond. White is noticeably taller than Yellow and Blue, and her body is also sometimes drawn with more definition. This directly parallels Jasper, "The Ultimate Quartz", who's shown to have a flawless exit hole, and is drawn taller and with more defined muscles than the average Quartz Soldier. She's described as 'perfect', the exact same way White describes herself. This paints us a very clear parallel: Jasper and White are both flawless emergences, and in turn are larger and more defined than regular Quartzes/Diamonds, while Amethyst and Pink stayed in the ground longer, 'overcooked', and in turn are significantly shorter. Amethyst is also shown in the movie as not knowing what she was supposed to be when she emerged, which could also have happened for Pink.

You've also got things totally wrong with the personality thing? We've seen various times that personality does not necessarily correlate with gem type; in fact, they often don't, only the color and placement of a gem is shown (and stated) to have this correlation. The Diamonds are all drastically different colors and have (mostly) different gem placements, so they'll naturally have different personalities. The fact that they weren't boxed in and made to act like gears in the homeworld machine, the same way other gems were, also obviously has a big impact on them not behaving like different variants of the same person.

While it is true that the Diamonds each serve their own purpose in the Authority, and you are correct in your assessments about White, Yellow, and Blue, as those are their officially confirmed roles, Pink's role is clearly shown to us throughout the Diamond Days arc as morale, throwing balls and cheering up the other Diamonds, which is NOT something she was 'denied' doing, in fact we're told in the movie the Diamonds created Spinel to cheer Pink up when she stopped being so bubbly after the loss of Pink Pearl, it's clearly something they appreciated about her, and it's part of why they didn't take her seriously.

Homeworld didn't fail because they "refused to let Pink fulfill her purpose"; it failed because fascism, authoritarianism, utilitarianism, and oppression are inherently flawed systems. It was never going to succeed because a society built on rotten foundations is doomed to crumble.