My teeter toter point is that the wheels are bolted on, thus not relying on gravity alone to maintain their position. The small amount of surface area that is touching the jack head is tiny and inherently unstable. Plus, the issue was going to Sam's and changing all tires at once, presumably not attached to the TV, but likely with the tongue jack down, but not affixed to Mother Earth. And if the tire guy at Sams takes off both wheels, there's nothing to chock. If the trailer is attached to the TV and its parking brake is set and wheels chocked, then it's a horse of a different color and the instability risk is vastly reduced, except maybe in an earthquake.

As for dealing with tire changes on the road, we carry an inexpensive torque wrench and socket so we can regularly check the lug nuts and it is easy to use to change a tire.