Well done, Dave.
Creating precision current sources is always a sort of Fine Arts.
The T.C. of the Vishay shunt resistor can be estimated from your video, when the output did not oscillate any more.
The first stable and maximum reading from the 34461A at 20:57 is 1.000635A, 1.000677A, decreasing due to heating of the shunt, stabilizing to 1.000627A at 21:30. (In first order, I assume that the 34461A does not drift.)
Therefore, the Vishay shunt drifts by 50ppm when loaded with 1.25W.
Specification states 4ppm/W, that relates to the "typical" +/-0.2ppm/K (-55..125°C) in the parameter table 1 in the datasheet.
Therefore, it should typically drift 5ppm only, but the real drift obviously is 10 times that value.
That means, the real T.C. is about 2ppm/K, still within the spread of the specification.
Let's calculate that a little bit differently.
Somewhere in the video, you estimated a shunt temperature of 60°C, i.e. a 40°C temperature rise.
That would give a smaller T.C. of 50ppm/40K, around 1ppm/K.
So, these Vishay shunt are really excellent, but their blatant advertising in the datasheet, i.e. printed in fat letters: typ. 0.05ppm/K @ R.T., is still a little bit exaggerated, by a factor of 20.
Frank