The "physics engine" in this situation is really just a "collision engine", and it doesn't impose any restrictions on you. Kinematic bodies just give you a convenient way to check for collisions between two points. You can use an Area2D if you want, which will do nothing by itself and only give you collision results every fixed step. Obviously, if you don't want the engine to do anything for you, then you have to figure out everything yourself.
What exactly do you mean by "wonky"? If you're trying to make an object behave in physically-incorrect ways, that's wonky. Super Meatboy's mechanics for example, are extremely wonky. You stop moving upward the instant you release the jump key, giving you a weird cut-off arc like you ran into a ceiling. Likewise you stop moving on the ground instantly if you release all the keys, but you slide like crazy if you try to reverse direction. Extremely wonky.
What kind of platformer are you trying to make? You mentioned tile-based movement and "integer based actors position", which means something like Flashback of Prince of Persia, where you can only move in full tile increments. That's completely different from Mario or Meatboy, where you have continuous control and can stand overlapping tiles however you want.
[Edit] Also, for fairness's sake, most of this has nothing to do with Godot. Physics engines all work roughly the same way, and game engines are all pretty similar too.
