Interesting video. Hmmmm...just throwing this out there, so bear with me. The tank was fine before, flow was normal then problem with well and the problem began. So now it appears that you are waiting for water and when it comes out it spurts and is cloudy. The behavior you faucet is showing, at first, reminds me of the symptoms of a leak on the supply line, maybe in a concealed area. I know it brings the question of how air would get in the pipes, but it does happen...dealt with it many times, but the sputtering usually goes away in a few seconds.
So, is there a possibility that the elevated tank is allowing water to siphon through the dip tube into the cold supply lowering the water level in the tank, even if it's only a gallon or two. When the faucets are turned on, the hot water and air are pushed out together as the cold fills the tank void at the top. This would have to occur in conjunction with a tiny leak on the hot side or a drippy faucet on the hot side to allow for the siphoning and air to enter. Your tank should have an anti siphon hole in the dip tube, but if it does not or if it got plugged with sediment, you could siphon an elevated tank. If the water is left to run for a few minutes, does the sputtering go away?
That's just a theory, but I'll keep trying to brainstorm on this one.
Did that make any sense to anybody other than me?