Jacking the axle up doesn’t change the position/state of the suspension, it will simply lift the whole vehicle up. The short version is, if you’re simply replacing the panhard bushings, all you need to do is park the truck, remove the bar, put new bushings in, and reinstall the bar. After reinstalling one end of the bar, you might need help shifting things to get the second bolt aligned, if the vehicle shifted at all when the bar was removed.
The longer version starts with saying I’m not an expert on anything, but as you can see in the first video you posted, he doesn’t jack anything up. With the bar removed, the truck is still sitting on the coil springs, with the axle held in place by the radius arms. You wouldn’t want to drive it without the panhard installed, but it should be fine sitting there while you get new bushings.
The following is irrelevant to your situation as you’re just replacing bushings, but to explain what he’s doing... an adjustable-length panhard bar is used to center the axle from side to side, after you lift a vehicle with solid axles. The lift changes the angle of the bar, and thus shifts the axle toward one side, as you can understand from the video I posted. He’s not adding a lift in the video (it’s already lifted), just installing an adjustable bar to move the axle back toward center.
If you look at your front axle and the bar, you can imagine how lengthening the bar would push the axle toward the passenger side, while shortening the bar would pull it toward the driver’s side, hence using an adjustable bar.
He left wheels on the ground, without jacking anything up, so that the vehicle sits at its natural ride-height (suspension is neither compressed nor drooping) - this is so when he installs the new adjustable bar and changes its length to center the axle, it will be centered at normal ride height.