You got to loosen up. And I wouldn't tap the fingers - play from the surface.
Playing fast is a skill of the mind rather than the fingers. If you can tap your fingers and making a percussive rumbling sound with your fingers then you already have all the velcity and finger speed to play like Art Tatum already. In order to play fast music you have to be well ordinated and able to follow the music very well mental;y. Go over the piece in your mind and learn the piece well enough you can hear the piece in your head. You should be able to anticipate upcoming patterns changes, skips, leaps and rhythm changes. Once you can comprehend all the notes mentally, you can work on dexterity by practicing hands separately, up to speed, with muscle relaxed, fingers close to keys, with the goal of eventually getting to end of piece with out pain to increase endurance. Practice using uneven rhythms ( long short, or short long patterns) to help keep the runs evens. Do the same with the left hand and then you can drop in the Lh chords in rhythm section by sections until you can do the whole piece. If you are really systematic with your practice, you can achieve some very quick results. But keep in mind if you cannot say the rhythm or audiate the sounds in your mind, you probably will not reach your maximum speed ....Hope that helps...
I'd say work at it from two ends. Spend some time practicing relatively slowly, for evenness and precision. But don't try to speed up from there. Instead, also spend some time trying to play them blindingly fast. Don't worry about missed or uneven notes, just pay attention to how your hand moves. You'll notice that you need to do little wrist rotations and move your hand towards or away from the black keys. When you play very, very fast, you'll perceive the pattern of these hand and arm movements more easily. Then slow down again, focusing on precision and evenness, but keep paying attention to the larger patterns of how your hand had to move at high speed. Eventually you'll be able to play both fast and accurately.