Mechanically speaking, I think wheelbarrows make the most sense as a vehicle.
Moving it around by the furniture hauling mechanic would be appallingly terrible (shift-8 > f > target > slide > move… that’s six keys! Every single tile!), and not at all representative of a labour-saving method. If, for some perverse reason, you wanted to transport goods around by pushing a dresser down the highway, that’d be an appropriate modelling of the amount of effort, but wheelbarrows are much easier than that.
On the other hand, representing it as clothing is both way too permissive (moving a wheelbarrow takes a lot more of your concentration and effort than something you can just wear and forget… to start with, it occupies both of your hands, much like a steering wheel would), and way too limited (not only can a wheelbarrow hold an awful lot more than backpacks, it also helps to manage the weight, and clothing can only represent the former).
In terms of the vehicle representation, sure, it’s a little weird that you’d need the driving skill to push a wheelbarrow, but the same could be said for a bicycle. Furthermore, the handling issues (wobbling, etc) that you get from vehicles model wheelbarrows pretty effectively, as anyone who’s used one before will know (it’s hard to keep a one-wheeled vehicle on track properly). So yeah, I think you do in fact have to represent wheelbarrows as a vehicle, if you implement them at all.