The way the story of Sonic 06 is split between the 3 hedgehogs is like, ...
Sonic the Hedgehog's story centers around the current events happening in the present, Dr. Eggman capturing Princess Elise III in order to unlock the power of the Flames of Disaster in order to gain control of time, and Sonic trying to save her and the two grow close in the process.
Shadow the Hedgehog's story center around the origins of the current events tracing back 10 years in the past, as he has to face his own shadow and inner-demons in the form of the corrupt consciousness of Solaris, Mephiles the Dark, and prove once and for all that he will never turn back.
Silver the Hedgehog's story centers around trying to prevent and undo the outcome of the current events that will result in the apocalypse 200 years in the future, as he trravels through time to in order to find a way to defeat Iblis the Flames of Disaster once and for all.
To make Sonic's story appear more important and revelant, you just have to expand on the fact that Sonic's story is the center of the entire narrative. It's the PRESENT. Shadow deals with past mistakes and Silver deals with future outcomes, but Sonic is in the HERE AND NOW. The Flames of Disaster that will burn away at the world in the future is currently sealed within Elise, and Eggman is trying to unleash it. Sonic and Elise are supposed to grow closer and closer through the story, so that should be expanded on, but most importantly, the fact that the flames are inside Elise should be vastly expanded on and maybe a little tweaked so that it comes into play and Iblis is nearly awakened a few times during moments when Elise is very emotional. Like, maybe Elise should be like a pyrokinetic with no control over their flames at all during a few parts of the story. Like, have Eggman actually try to unleash the flames from Elise instead of just talking about it all the time.
Shadow's story has the bond between Team Dark further developed, and Silver's story sort of develops Silver and Blaze's teamwork and stuff, and even Amy has a few character-centric moments. So in Sonic's story, Tails and Knuckles should have their own character arcs in relation to Sonic and stuff as well. Like, for example, have Tails' independence further developed to show that he's truly growing into being Sonic's equal partner and no longer just a sidekick, he's becoming more of a hero in his own right and doing things on his own and being someone Sonic can rely on. For Knuckles, uh... hmm... maybe have the reason he came to Soleanna be because of the Master Emerald acting strange and responding to something down in the city. Like, the Master Emerald was ... "resonating?" with the power of Solaris and the space-time anomalies that are occuring, and Knuckles came down to investigate. Maybe make use of his role as Sonic's Friendly Rival and have him 'compete' with Sonic to some degree while also helping him stop Eggman and save Elise.
That's all I got for now.