What’s your stance? Should a guy operating a lawnmower on public roads while drunk get charged with a DUI?
No. Maybe drunk in public, or some other similar thing.
I look at it this way: A lawnmower is not a motor vehicle by definition, as it cannot be legally registered and operated on a highway - some states also require recreational vehicles (ATVs, Dirt bikes, snowmobiles, etc...) to have registration stickers for use on public land. Lawnmowers don't fit this category, either.
It may fall under a similar category as a farm implement that may be operated in short duration on the highway between fields. If there is a rule against that, or a rule against operating unregistered, self-propelled devices on the highway, then maybe.