I understand it is a radish and it helps break up the hard soil. I was thinking of trying it next year.

I was leaving the posting to those with a more appropriate climate but...
I mainly grow daikon to eat (makes great pickles and kimchee), but it grows like mad and I always end up watching some get pretty huge.
I'm in sandy soil, so breaking it up isn't an issue, but they certainly would!
In temperate climates like mine, it's generally recommended to chop large roots off at ground-level and leave to rot in the ground.
I presume in your climate you just let the cold do the work!
My only concern would be a brassica cover-crop mucking up rotation.
I don't rotate as my garden's tiny, but it could be worth factoring in.

I've used radish and that family (turnips and the like) for cover-cropping here in zone 5, and they don't reliably winter-kill. And then my mother (who is originally from Japan) harvests the leaves to make pickles with. But unless them overwintering is a deal-breaker, there's really not a downside. I'd much rather deal with trying to weed radishes and turnips before planting than tough grasses, ground ivy, or curly dock.

Buckwheat is supposed to dig down well, too. It's too late to get that established this year, but in future years that's a possibility. And that *WILL* winter kill for sure.