Interesting question for a tagalong bike.
QR skewers are generally fitted to vertical dropouts so all they have to do is hold the wheel in place if it loses contact with the ground. If we consider what happens when I apply the brakes so hard that I skid (on my vertical dropouts), the wheel effectively gets locked at the brake pads, the bottom of the wheel is pushed backwards (against my direction of motion). The axle is another "lock" point, my axle presses against my frame and thus we have two lock points and the wheel stays well and truly attached to my bike.
There is/was a problem when QR skewers started being used with disk brakes. Imagine placing a disk brake pad exactly level with a vertical dropout and then imagine applying so much braking force that the wheel effectively "locks" at the brake pad. The wheel would also be pulled backwards under braking. Imagine your bike and rider weigh very little and your QR is loose - can you see the problem? The frame might lift away from the axle (which is only held to the frame with the not very tight QR). This did actually happen, which leads me to conclude that the QR cannot be relied upon to counter all forces exerted on a wheel.
As such, I would be wary of holding my axle with only a QR on horizontal dropouts - not only does the axle need to support my weight (obviously) but there's a whole host of other forces pushing the axle forward and back during bike movement, forces that want to make the axle slide about the dropout.
Going over potholes etc would cause forces in the direction of the dropouts; this might loosen the QRs over time, who knows. They'd quite possibly cause the axle to move about the dropouts, given enough time.
I would
slime the innertube instead and carry a pump *waits for torrent of abuse for suggesting slime*
