I think Knuckles best plays how he does in all the games he stars in. He glides, he punches, he gets to areas other characters can't through rock or the air. He can dig, he can even sometimes swim.
What you've described about your ideal Knuckles gameplay is kinda what his style already is. I disagree entirely that he shouldn't be able to run fast as often- because he can and he should. If you're going to redesign his gameplay, perhaps put a few more climbing sections in which require skill to manouvre, and maybe put some more secret areas in with rock covering etc. Apart from that I can't really say his gameplay hasn't always been suited to him, because every game I play of Knuckles- even treasure hunting- revolved around a similar premise and played well.