66 Mustang 289 C4 Jumps into Reverse after Starting.

One of the main two house rules at my Shop as it has to do with these ancient Mustangs is, If the engine is running, someone is in the drivers seat. I've had two C-4 automatics drop into gear on me in the past. The first time years back while out delivering phone books in the winter and left the engine running to keep the heater going, got out of the '66, walked up to someone's porch to drop the book, turned around and it dropped into reverse. Fortunately I had left the drivers door open and it took me about 75 feet to run it down and jump in the seat to stop it. It stopped just feet from a large brick mailbox and fortunately no one was passing down the street that it crossed. It happened to me again with a different 'stang one cold morning at home. I had started the engine, realized that I forgot my winter hat, left it running to run in the house, came out it was backing up towards the pasture fence. I ran it down too. (I was a much faster man on my feet two decades ago). I had to open the door this time and jump in. A similar thing happened to our Mustang club president after an inexperienced with old Mustangs repair shop had his C-4 out and back in. It drove itself out of his garage and backed into a tree. It fouled up the taillamp panel, bumper and trunk lid. After this happened we made a rule at the Mustang show that if someone had their engine running then someone had to be in the drivers seat. Now I jack up the backend and put it on stands if I am going to be working under the hood and periodically starting it
The other main rule that we have is we don't leave battery chargers unattended especially overnight. How do you sue China if a battery charger burns your car down?
We also have rules having to do with gasoline but I won't get into all of that.
Be careful. These vehicles can hurt someone if one is not in tune with everything going on while repairing them.

As far as the little ball on top of the shifter goes, if you can move it with your thumb and the little gate? moves up and down in the notches then that cable is in good shape.(Edited-I see now after re-reading that you said it was good). The black push button in the left side of the T-handle can easily be installed the wrong way. It will only work one way. As far as the "kickdown" cable goes, there are two ways that the arm that it is hooked to the throttle pedal assembly. Some were welded, some had a very small U-bolt attaching the arm to the throttle rod. A welded one could be bent out of adjustment and the U-bolt type could be loose or tightened down in the wrong position. Maybe a user here has the info on adjusting that, that they could share. If I was standing over it I could adjust it. However the basics of it are, it shouldn't have any tension on it until you press the pedal to the floor.
I remembered a 3rd situation where one dropped in reverse on me. It was V-8 powered and The coil wire was in reach to yank and shut down the engine. If it had been straight six, the coil wire would have been harder to reach!
EDITED-I just watched your good video and your shifter handle cable is good.