I saw a guy that hauled 2 karts on the bed of his pickup. He used 2x4's and plywood to set across the top of his box. The karts sat sideways on top of the box. That doesn't help much for the lone guy trying to load and unload.
The kart stand you show in the video looks like a handy device, but I don't think it would work at tracks like where my son races. The pits are very rough. The pits are not packed dirt, it is grass with a lot of gopher holes. The stand in your video would be too wobbly at the top and the front wheels would get stuck easily. I made a stand our of some old square tubing I had laying around with some 6 inch lawn mower tires. I made it so it can come apart quick and easy for storage. Again, that would not help you because it is a 2 person operation.
I would suggest taking a close look at your pits to see how easy it will be to push a kart, then look around for some of the stands that use electric winches or boat winches. Whatever stand you choose just keep in mind how easily it will roll and how stable it will be pushing across the pits.
How tall is your pickup? I am able to load and unload my sons kart into my pickup alone by lifting the front onto the tail gate then lifting the back and shoving it in. He has to help steer it but lifting is not too hard doing it that way. Our kart stand is tall enough I can pull it out of the pickup right onto the stand. I do have a little 4x8 home made trailer that we use when we haul it to the track with our expedition.