You asked - "how important is genre?"
I'll answer your question with a question: "How important is genre TO WHOM?"
To me, not at all. I'm not capable of distinguishing from one genre to the next musically.
To a songwriter who is also a musician competent enough to be trying to get on the radio - it's vital.
To a producer trying to help create a sound that will sell for an artist - it's vital.
To an agent, trying to sell the package (artist/songwriter/producer) to a label, it's vital.
To the label, it's vital. . . or not. Sometimes genre bending works. But it's almost certainly best not to bet that it will unless you can afford to lose the bet.