With his Wind Scar, InuYasha can defeat just about anything. With a 'sit', she can defeat InuYasha. But in a heated battle, one where the Wind Scar won't work, Kagome is incompetent. She has her arrows, but what happens when all of the purity is gone? Her arrows are pure, but will they really last forever? All she can do is detect shards. That isn't use; that's just a stinky advantage over the enemy. She could die, and not much would happen. Kagome is just plain incompetent.
The Tetsusaiga is all powerful. InuYasha's father was all powerful. Naraku is all powerful. InuYasha is not. He may be all powerful over demons, but to his inner feelings, he is equivalent to a worm… or even Shippo. He hates feelings. They make him feel vulnerable, and ergo he feels incompetent. He can't even make safe decisions about Kagome. Every decision he makes, Kagome is hurt. He is useless. If he just died, Kagome could just go back home and be safe…. She could be safe, if he just left. But InuYasha needs her. Not just because she detects jewel shards, but because… he, InuYasha is incompetent.
All he does are illusions. The simplest being can see through an illusion. What makes him so special? His big, stupid illusion of a big, pink, circle-looking thing? His little donkey? No, there is nothing but incompetence. He has no fighting ability, no sword; he is too puny; he is too whiny; he is too annoying, and the list goes on. If he died, the only one who would miss him would be Kagome, but she has InuYasha--she'll get over it. No one will remember Shippo or his incompetence.
He is perverted, and that's not getting him anywhere. His is cheap, and that is not getting him anywhere. The only thing keeping him alive is what's going to kill him in the long run--his wind tunnel. Some day he will die, and people might rejoice. No more, "Will you bear y child?" No more. His incompetence to help InuYasha except for exorcisms will be gone. No more feel housing with the occasional powerful demon. No more. Hell, he even thinks Sango will leap to her feet in joy. One day, the wind tunnel will consume him, and Miroku's incompetence will be gone.
She is a woman. Therefore, she might be classified as incompetent even though she is a warrior, with a boomerang bone and Kirara, who gets them anywhere countless number of times. But most of the time, she's hanging back because the men have it under control. Sometimes she would rather be a man rather than be an incompetent woman. She can fight, so why doesn't she? Because she knows that she has to protect Kagome and Shippo and those who are weak. Sango can't help but feel bad when she wants to fight and would rather leave the young, innocent, and helpless for dead. If she were to die, she wouldn't have to worry about them. But will Sango really let her incompetence overshadow her desires?