The Master Emerald was introduced in Sonic 3 as the main objective of Eggman. Being more powerful than the Chaos Emeralds, he could power an entire space station with it alone. Angel Island was held in the sky by the power of this emerald, and it was guarded by the echidna named Knuckles. Where Sonic 3 ends, though, is a huge load of mysteries for players to solve. Why was the Master Emerald on Angel Island, why was Knuckles the only one there to protect it, and what is an echidna (Goodle it, they're real)? Here's what I got out of it.
Sonic Adventure gave us the backstory of Knuckles and the Master Emerald. Apparently, he is the last of his kind and the Master Emerald turns out to not only be the most potent power source on the planet but also the prison of an ancient, vengeful creature. When the Echidnas of ancient times got greedy and seized the Master Emerald from the Chao and their guardian, Chaos, they were wiped out. Chaos became untrusting and vengeful of the echidnas and likely the world, so to stop his rage he was sealed inside the Master Emerald. Eggman learns this (to an extent) and breaks Chaos out. Angel island falls, Knuckles has failed to protect it, and the ancient evil is once again running loose.
Eggman's end of this story is also something to note. He is launching his biggest campaign (at the time) in this game. He has a mobile flying fortress called the Egg Carrier, the start of a robotic E-Series Army, an extremely angry and powerful being working for him (he believes), and multiple backup plans to try and ensure his victory. Egg Carrier crashes? He has a base on standby. Chaos is defeated? He has a nuke on standby to wipe out the town and make a clean getaway. Chaos defies him? He has a specially armed Egg Carrier on standby to take him out. Sure, none of these backups actually work, but that's a big step ahead of his norm up to this point. He actually shows real evil when he plans on nuking an entire city. Eggman was at his finest during the Adventure era.
So with Knuckles and Eggman taken care of, let's talk about how this wraps up the Sonic series (pretending it were the last game). Knuckles redeems himself by restoring the Master Emerald to its full form and returns Angel Island to the sky. He didn't need Sonic's help like he did in Sonic 3, proving he is good enough to get the job done on his own. Where his story goes from past this game is irrelevant (seriously, his contributions continue to minimize every game). The Master Emerald may still have all the power it had before when Chaos was still sealed, but he no longer had to worry about Chaos. The purpose behind him guarding it was completely gone and replaced with just taking up the mantle Chaos and the Echidna had left him as the Emerald's guardian.
Tails is broken away from Sonic for a while and finds himself in it alone. He deals with the separation anxiety and stress of having to rely on himself to save an entire city. Sonic isn't there to help or guide him, and Tails goes off on his own knowing that he has to be the hero. The gameplay of Tails further demonstrates this as the entire game long beating his levels means getting to the end BEFORE SONIC. Tails in the end manages to save the city of Station Square from Eggman without the assistance of Sonic.
In the end, Sonic is forced to transform using the positive energy of the emeralds (corny, cheesy, sappy, etc.) which is something only he can do (he seems to be the only one not influenced by negativity). He chooses to take the challenging road to fight Chaos and try to show him that there are still some left with a good heart. Chaos ultimately loses and sees that the Chao are still alive (the victims of the echidnas trying to steal the Master Emerald). Tikal, the one who sealed him, makes peace and they leave together. The ending of the game sees Sonic following Eggman out of the city.
So here it is in a nutshell. Tails overcame the fear of being a lone hero and made a name for himself. Eggman finally put his trust in a creature that was not of his own design and, when pulling out all the stops, still couldn't stop Sonic and friends. Knuckles discovered the roots to his culture and duties and also left the comfort of Angel Island on a successful mission to restore the Master Emerald and vindicate himself. Sonic settled a dispute between two ancient beings using his positive influence and brought them together as friends in the end. Yes it means we would have no Sonic games to play anymore or the less story-driven games we have today would have come sooner and characters like Shadow wouldn't exist, but from a story perspective it would have ended the series on a high note. Eggman wouldn't become a repeat offender of putting his faith in things he can't control and characters would stop showing up as one-offs.
I'm not hating on any Sonic games in this blog (I did plenty of that on gamefaqs when Generations came out) because the different games appeal to different people. I found Generations to be enjoyable to a point (much better than anything released since Heroes) and like I said before, I'm basing this off story elements.