Настя, а где вы увидели "могу"?!

This means “You best let ME deal with this” (=I’ve done it before/I know what I’m doing/You’re not doing it right/You’re in no condition)

? А почему бы МНЕ этим не заняться?

Imagine a woman who has just found out her husband died in an accident at work. Her neighbors find out and go straight to her house. When they walk in, they hear crashes in the kitchen. The dishwasher is spraying water everywhere and the woman is in tears, hysterically taking out dishes and throwing them on the floor. The man runs over, gently takes her by the shoulders, smiles reassuringly and says “Why don’t I deal with this. You go sit with Angie in the living room and I’ll take you a cup of tea.”

Ну тогда просто, без "А" - Почему бы МНЕ этим не заняться?

Как насчет компании, в которой кто-то делает чтот-то очень плохо. Ты мог бы это делать лучше и хотел бы, но тебе никто не поручает. Тогда ты говоришь начальнику с элементом недовольства/раздражения - А почему бы МНЕ этим не заняться?

In that case you wouldn’t use DEAL WITH - you would say “Why don’t you let ME do that?”

DEAL WITH is used to refer to a problem, crisis that needs to be resolved - especially if you’re doing it for/instead of someone else. “When Frank died, my children dealt with all the arrangements and legalities because I was a zombie.”

✅

+++Grumbler