While you should definitely exploit the idea of more villains being on the rise(this franchise desperately needs more villains that actually stick around), one question the audience would most definitely ask: "Why have we never heard of the other evil geniuses before?"
These guys either can't be too big and loud or they need their own archenemies who always stop them like Sonic does with Eggman, so they never became a major problem before. I actually kind of like the idea of there being more villains and heroes than just Eggman and Sonic(who just happen to be the greatest of them all) which is one of the main reasons why I think a Team Chaotix spin off would work so well. Have some of the other bad guys work solely underground like in the mafia or make them shady government/business men who are considered good guys by the public or make one of them a king of thieves who only works in secret with his minions, so there's no way most people would know they even exist. This would make for a lot of variety considering the abilities, boss battles and motivations of these characters which is important since they and their interactions with Eggman would have to be really interesting and entertaining. Some villains might like working for Quark as they benefit from it while others hate it and only fight Eggman because they're forced to. Quark would have to be a big deal though, so making him the only other evil inventor besides Eggman(and maybe "Omelette" too, but she'd probably just be a beginner) would be only logical. I'd actually say that he would need an archenemy of his own to parallel Eggman even more. Maybe even have Quark show off by having him actually capture and brainwash/roboticize his own archenemy while Eggman can't even lay a (permanent) scratch on Sonic. Said archenemy could then later be "rescued" by Eggman and drag Quark to jail in the end. I don't know if you planned on doing anything with D.U.F.U.S., but I'd say make him more threatening and have him always be around Quark for sometimes more, sometimes less comedic purposes. Quark could go more over the top without losing his intimidation when he has a sidekick like this. I'd say something like Fawful and Midbus from Bowser's Inside Story or Aquamarine and Topaz from Steven Universe would be a fitting comparison. I don't know about the thing with Omelette, I'd say don't have Eggman adopt her because there's no way this would actually stick outside of this spin off game/series so it would be near meaningless, but instead have him appoint her villain rookie and give her all of Quark's remaining inventions after Quark is defeated. This would be heartwarming in a way considering Omelette is usually portrayed as somewhat of an Eggman fangirl, but it would also be a logical thing for Eggman to do as to him, the robots are in better hands with her because she won't use them against him, so he'd still be somewhat following his own interests. Maybe even have Quark dismiss her before and say that she's just Eggman's only fan or something which would make the ending even more painfully ironic for him. Throughout the game, she could be the tutorial fairy who explains to the player how things work and helps Eggman out all the time like giving him hints on which robots and weapons to use against certain opponents.
Of course, it would all really depend on the tone. If it's more comedic and cartoony(which definitely is an option considering the show Dr. Quark is from), the other bad guys would be more card carrying villains who use "evil" as some kind of fashion and the overall plot wouldn't be too serious. That would allow you to get away with more and exploit some crazy ideas, but also would make the whole thing lose weight. Though it wouldn't be impossible to pull off, comedic cartoony Sonic has always been lacking in the organic character department and heartwarming moments which I'm guessing you'd want to go for with Omelette.
If you plan to play it more straight on the other hand, you'd probably be able to make the characters more organic more easily(no matter if they're actually organic or not) and you'd be able to be more clever in terms of world building and events. Don't forget that Eggman is a bad guy after all, a terrorist even. You can't have him get reformed or even soften up too much, it would go against his character. The second option would allow you to play his and the other bad guys' villainy more straight. Eggman returns to the underground after fighting Sonic just to find that Quark has seized control of most of it and made contracts with other villains so they'll work with him. Eggman is pissed and does whatever it takes to reinstall his law and order into the underground once again. To do that, he has to go up against his collegues until he reaches Quark. I'd probably prefer the latter option, but there would obviously still be a lot of comedy. I just think it's important that Eggman is an actual threatening villain in this game, even if he's the protagonist.
As for the gameplay, I'd say concept-wise, do something similar to the Minions Quest in Mario&Luigi: Superstar Saga + Bowser's Minions where you can recruit all kinds of enemy monsters from all kinds of Mario games into your army but in the Eggman game, you'd recruit Motobugs and Buzz Bombers and all the other robots Eggman commonly uses with the likes of Metal Sonic and the E-Series joining in later on. Have Eggman collect tools and materials to build new robots or upgrade old robots and have him use them during gameplay. The Pikmin thing doesn't sound too bad as it would be a way to have Eggman keep his robots around, but I personally prefer it when games are more interactive than Pikmin or the Minions Quest are. You should be able to manipulate the AI of your army so they'll pull off certain moves, follow different strategies, adapt to the current situation etc. Eggman himself should control very well and definitely not be defenseless. I'd say have him upgrade his Egg Mobile/Walker as well so he can pull off different moves and attacks with it, the idea of making individual vehicles sounds good too, but it would be rather hard to pull off properly. Make it kind of a fighting/puzzle game where you blast you opponents in a grand fashion while you also put your brain to use. This would fit Eggman's personality extremely well as he loves to be big and loud, but he's also very smart.
You know, now that I think about it, I really like this idea. Screw Blaze, this is my favorite concept of yours now. Rebooting Dr. Quark as the big bad for this game is a great idea and how this whole rebooting thing should be done as it is taking a somewhat familliar villain to a whole new modern level. There's a lot you can do with the underground and other villains as well and Eggman building his own machines to show them who's boss could lead to some pretty great character moments. I like the idea of a character like Omelette being Eggman's sidekick as it breaks away from all the tired sidekick clichés Eggman was subjected to in the past decades. You should definitely exploit this idea more.