Gosh okay, I have literally been VIBRATING all day to write this so here it goes because this has LAYERS of analysis as to why this works.
The first point requires some background on Romeo and Juliet. The reason this monologue moment works so well has a lot to do with the play itself. Romeo and Juliet is ostensibly a romance on the surface, but if you look a bit deeper past the star crossed lovers it's a tragedy. Shakespear hammers home the tragedy of the play a lot in the script, going so far as to make sure there is an opening AND closing monologue about how sad it is.
These short prologue and epilogue set the stage and act as a final reminder to the audience that this story is not about to lovers, but rather their death and the circumstances of their deaths. The play is, first and foremost, a tragedy. And this monologue is the true start of the tragedy of Romeo and Juliet. It's the moment when things spiral out of control for both title characters. It's the moment when Romeo firsts gives in and acknowledges that pursuing Juliet is a bad idea.
"Arise fair sun, and kill thy envious moon, who is now already sick and pale with grief."
These words are powerful, and paint the moon (Romeo) as heart sick for something that cannot be. The word "kill" invokes a violent end, foreshadowing how things will end for the doomed teens. It's also worth noting that Romeo falls out of the iambic pentameter during the short monologue, indicating his mental state (but I'll come back to this point in a second so remember it).
Now, on to why this is so freaking brilliant and works so well. It comes down to three points: the sun and moon imagery, the incorporation of Hyrule as the speaker, and the actions being done in the scene.
So, the sun and moon imagery are rather obvious in the monologue, and it lends itself REALLY WELL to both Twilight and Time. Twilight is associated with the sunset and the sunrise, and there is also the homophone play on words with "sun" and "son" since Twilight is Times descendant. Time is notably connected to the moon thanks to Majora's Mask, and the description of "sick and pale with grief" is HIGHLY applicable to a man who is watching their child most likely die before their eyes. The metaphor fits almost TOO well.
This use of imagery on its own to describe Time and Twilight is beautiful and amazing, and is enough to provoke a quiet "oh shiiiiiit" out if most of us but there is MORE.
Making Hyrule the speaker is a powerful choice because it separates the story somewhat from both Time and Twilight. Yes, the monologue and the imagery is *about* them, but it's no longer autobiographical. It takes lines that were written to inform the audience (even the ones sitting in the cheep seats where seeing what was going on may have been a bit... difficult) that Romeo was talking about Juliet who just walked on stage. Having Hyrule deliver the lines keeps the story about Twilight and Time but takes away the stage direction element of the original monologue.
In addition to this, the changing of Juliet's name to Twilight resolves the iambic pentameter in the monologue (when you add in the excluded "But soft," that starts the monologue). Why is this a big deal? Iambic pentameter is a nifty way to tell the mental state of a Shakespeare character, do they fit in with the pattern of the poem, or have their emotions gotten so strong that it is impacting them somehow. With Romeo, his emotions have overcome him (and it happens on Juliet's name no less, a not so subtle indicator of what has caused his emotional upheaval). Juliet's name is just one syllable too long to fit the pattern, however Twilight only has two syllables to his name.
This resolution of the iambic pentameter takes lines that once we're for an emotionally unstable character, and makes them into words spoken by a level headed party. It's distinctly Hyrule, who we know from the update doesn't truly get emotional untill the healing attempt has failed and everyone around faer is trying to console faer for failing. Which is YET AGAIN appropriate as it shifts the breaking of the iambic pentameter to "Arise, fair sun, and kill the envious moon". The emotional snap, if you will, now falls on "moon", or if yall will stick with the metaphor, Time. And it was Time who reached out to Hyrule to as of to say "you did your best", Time who Hyrule snaps at in the update "giving up".
Now add in the physical poses chosen for Mints short little edit and OH BOY. The first image has Hyrule reaching, gesturing both at faerself and to Twilight. The second image, where Hyrule looks down at Twilights failed healing with fists clenched makes the line "Arise, fair sun" into an order. And finally, the shot of Time, with Hyrule speaking over it brings back to mind Hyrules disgust and chastisement of Time at the very end of the update.
Taking it all four parts together (the sun and moon imagery, the choice of having Hyrule speak and with the single modification to the script, and lastly the actions being done in the scene, all set to the backdrop of one of the most iconic tragic stories of English history) and you essentially get a speed run storytelling of the update that captures Hyrules part of the story in it's entirety. From faers first reach to try and help, to the determination that fae will heal Twilight, and to his disgust with Time while STILL mantaining the initial theme that Shakespeare intended the monologue to have.