He's pretty wise, got great martial arts skills that help him go one on one with the Avengers with no rings. He is a monster at fighting and is a top scientist  he was able to mind control dragon Fin Fang Foom and had the biochemistry skills to engineer a virus that killed 97% of the Earth's population making 3% totally free from disease and immortal
 
With this kind of mad resume why would he need rings?
 
 

 He doesn't need them to be a threat, but he is cooler with them. They give him a cool visual that evokes pimps and goodfella thugs. 
Also, he uses them to compliment his martial arts attacks. Creating illusions and then sucker-punching the other guy while he is fighting things that aren't there. Freezing the other guy and punching him while he can't move. Tornadoing the other guy and sucker-punching him while he's dizzy. Etc, etc. He uses them the way the villains in martial arts movies use throwing dirt in the other guys eyes. Don't think of them as a separate attack. Think of them as a way for him to be a cheap, dirty s.o.b. while fighting using mainly martial arts.

The Mandarin's rings aren't a crutch, they're just a way of making sure that he can handle opponents who would otherwise be too much for even HIS skills to handle!

He's still able to trade blows with Iron Man level characters, so he's an immense threat. Even his son Temujin, after charging up his chi, was able to pressure point Hulk which made him scream out in pain. He's tanked hits from Thor, reacted to bullets from a few feet away, wrecked Hellcat and Black Widow at the same time in h2h, and has stomped Jubilee and Psylocke. That's not even getting into the crazy technology he has and his vast intellect along with it.