It's absolutely ESSENTIAL that there be a design focus on terrain navigation via momentum rather than the combat focus of a game like 06 and heroes (heroes did it well, 06 didn't). In the previous Adventure games, the core gameplay was built around getting to the end of the level as quickly as possible, using the terrain to your advantage in SA1 and optimizing your path in SA2. Playing SA1 well depend more on creativity with your movement options and SA2 is more about learning the best route and mastering it (for the core Sonic gameplay anyway). Compare that to 06 which has you wailing on groups of enemies to get from one gated section of level to another. All (or most in the case of the magic gloves) of Sonic's moves in SA1 and 2 are about traversing the levels. The bounce attack lets you reach higher ground at steep vertical inclines, the spindash lets you pick up speed quickly. Where combat exists, it's with singular, one hit enemies or bosses and enemies are primarily used as platforms with an element of risk (gotta time your attacks right so you don't get shocked mid air and fall). The homing attack itself has very little hang time, only enough so you can make decisions about your direction mid-air. Compare this to 06's abilities, where the bounce attack is basically never used for platforming, rather, its. primary function is to get consecutive hits on large enemies quickly. The spin dash in 06 is used to destroy small groups of weaker, single hit enemies quickly. Since levels are sectioned off in 06, it's not like having more traversal focused movement would be more helpful or better. You'd still be running up to groups of enemies with health bars (which is a stupid design decision for games like Sonic Adventure where you want to keep the player moving) and wailing on them until the door or box with a switch in it opened so you were allowed to get to the next part of the level.
For a hypothetical "Adventure 3" I'm going to base the core movement mechanics off of SA1 because it's the game which more closely resembles the classics on a mechanical level. I'm going to take the best elements of games that came after like the bounce attack and instant light speed dash (unlike SA1's light dash which needed charging). Sonic's momentum works as a gradient, going from a walk, to a jog, to a run, to a sprint, to top speed, represented with their own unique animations, top speed having some sort of animation smear for the legs. Sonic's general movement is very light and responsive. He can turn on a dime at slower speeds but it gets harder as you go faster. Rolling and momentum mechanics being as core to Sonic gameplay as they are, they'll work basically as they did in the classics. Sonic gains speed going downhill and loses it going downhill. That speed can be used to reach greater heights and find new places to explore be they level paths or collectible goodies. All movement mechanics operate on these physics. The homing attack doesn't simply stop you in place, your movement carries forward (further the faster you're going, higher the higher you hit the enemy or item capsule from). ENEMIES HAVE NO HEALTH BARS UNLESS THEY'RE BOSSES. This is self explanatory but to be clear, Sonic games are less fun when they force you to sit still, having to wail on enemies ensures you do so. I should mention, I'd considered having a crouch button like Sonic Utopia for rolling and the spindash to emulate the classics but this is a Sonic Adventure game. Single button spindash is how we're doing it. It shouldn't need to be said but omnidirectional 3d movement and an analog stick controlled camera are a must.
The levels will expand on some of the ideas present in Adventure (like exploration) while incorporating some of their own. There will be multiple paths to reach the goal but those paths are somewhat connected by areas you can explore to find items and upgrades. Something I've thought about is having multiple level exits a la Starfox 64 to add replayability but honestly I think the Chao garden has that covered (this is a Sonic Adventure game, OF COURSE there's a Chao garden).
There are multiple playable characters and I want to stress that they ALL have the same basic physics and rules. No treasure hunting levels or racing levels. Instead, goals like that will be relegated to missions a la SA2 and can be approached with any character. Sonic, Tails, and Knuckles are definitely playable. Amy, Shadow, and Rouge are LESS certain but I'd at least want Shadow and Rouge as Alt skins.
The visuals would be largely based on Yuji Uekawa's designs for the cast (he designed "modern", green eyes, Adventure Sonic), with the environmental design being something like a cross between the Adventure games', The Advance games, and Sonic Heroes. Still having that "classic" look while fitting more with Uekawa's Sonic designs. Character animation is to be done to maximise expressiveness in both facial animation AND movement animation. Watch SA2's Cutscenes to understand what I mean. In the Egg golem cutscene, you can see Sonic doing stretches in the background. Stuff like that.
The music will take cues from All games from this era, not just the Adventure games. Vocal character themes, some by Crush 40, Some by others. Level music needs to sound very SEGA and fit with the level theming and/or the character being used and/or the story moment the level takes place in.
The story will be more straightforward than past entries. Still full of high stakes (that the characters actually seem to CARE about) but not split off into a huge interweaving narrative like it's been done in the past. With 3 main characters, even if it goes that route, it would be much simpler than it had been. I don't want levels cut off just because a character's story doesn't take them there so we're sidestepping it altogether. The tone of the story is action movie serious. Comedy is allowed but it should be character based. Jokes shouldn't be funny because the JOKES were funny, they should be situational to the character. Sonic making puns and winking at the camera is a no. Sonic teasing of Shadow for being super serious while simultaneously wearing skates is a yes (as an example).