He's still recognizably Loki because he is still a trickster. They removed any power that would move him from scheming trickster to cosmic reality warper. Instead his few powers compliment him being scheming trickster without rendering his scheming and tricks superfluous.
The Mandarin is devious, savage, and exploitative. He doesn't need cosmic reality warping abilities to be savage, devious, and exploitative. The ability to chi-amp his body, genius intelligence, and a large organization of looters and conquerors ruling over slaves suffices for this.
If they do give him rings, it would be best to keep them fairly weak, essentially next-level versions of ninja smoke bombs and bolas tossed at the ankles and dirt tossed in the eyes. That's mostly how they were written in the Silver Ages, and keeping them at Silver Age levels works best for a character who is mostly about personal martial arts abilities, and brilliant organizational scheming.
For example, here is a typical Silver Age scene. The Mandarin creates a bunch of illusions of himself with his mind ring, and uses it to compliment his martial arts abilities. The scene obviously doesn't work if you amp the rings to the level where blending them with martial arts is superfluous.