That nice bottle jack with the stand, looks neat.
If you want to see if it works, and don't want to let the air out , I would not want as well,
Get a 4x4 or scraps or 2x4 and make a stand about the same height as the side wall of the tire.
Then put that stack under the truck frame and see if the jack will sit on it and clear the frame
I have not had my jack out, other than to check it and the winch on the truck since about 1998 in a blizzard at -25 with a load in the truck.
I do carry a big ass breaker bar and 2 good sockets, under the back seat.
I run winter rims and they use a special splined socket.
I don't think I could change the nuts with the factory handle as my hands have no strength in them any more.
And older trucks, check the spare tire winch , those can go to hell, if running in salt.
Re,the skid steer driver, I picked up one on the road yrs ago, walking down a country road.
He had blow tire up on the trailer, and no spare. I asked him why did you not just unhook the trailer , and drive back to town? He gave me that "look" , so I dove him back the 1/2 mile to his truck . I have had brain farts at times al well, but not that bad.