I have run open 12', 16', 20' trailers over the last 8 years. And I have the same question. I have thought about a box truck for the same reasons. The maintenance on trailer brakes, bearings, decking, tires etc is not cheap. I see several larger companies in my area with box trucks, and they sure as heck don't look as cool but hey, if it saves/makes you more money why not. Plus advertising space on the side. Fuel would be about the same depending. Plus it's enclosed. And you can still pull a trailer with it. I've seen them pulling skidsteers.
When you say you don't know what to do with debris, I assume, or at least for me spring thatch and fall leaf cleanup debris. Where to put it? You would need to pull a dump trailer with leaf loader and have a side door with ramp for the box truck. Or have the box truck be able to dump! And blow your debris into the truck. Then for fall cleanups pull the trailer with mowers.
I think the bigger companies around me have a second truck for debris. But I like my dumping box truck idea better!
2008 Toyota Tundra D/C 5.7 4X4
1991 Chev 3500 9ft Dump 5.7 4x4
20ft 14,000lb open trailer
M&G 6.5' x 14' trailer w/ 1' side
2007 Toro D590 60" 27HP Diesel w/triple bagger & dump front seat
2005 Exmark Lazer Z CT 52" w/ Trac Vac 655 Pro
1995 Toro Proline 52" w/Trac Vac 470 & TruTrac
Billy Goat 16HP Debris Loader
Little Wonder Edger