Personally, I went with the wire style because of the ease of cleaning. Urine, fur, pests, mites etc can't get into wire. I can powerwash, burn, sanitize, air out everything. I also built stackable ones, so I can utilize more space. Some I keep in the garage and some I keep in the yard, and am in the process of building a roofed leanto out of pallets to protect from elements. Downside is dumping the catch trays, but they give me lots of flexibility on where I put the cages.
I think your thoughts would definitely work, though I'm not sure about how to truly sanitize the wood portion. If I were, I would use a wire bottom for sure, and still install metal urine guards around the perimeter of the wood box portion to keep things a bit cleaner. I think I would use wire bottom in the box part too--you can always add a resting board if you want solid for winter, but it would help w ventilation. I would maybe go 24" deep (you don't want to try and reach further) by 24-30" on the run part. Maybe 12" for the box part. 18" tall.
Honestly, the other nice thing about using a nestbox is that it can be removed to check on kits, or even brought in during inclement weather if needed. If they just nest in the box portion of a hutch, you would have to transfer them.