In Thumb Position what works for me is a change in the action from "Mainly a Rolling Motion with a Sideways Component" to "Mainly a Sideways Motion with a Rolling Component".
Please bear with me.
If you sit at a table and rest your left hand on your finger ends as though you are playing a keyboard you will have your fingers arched (half bent?) and will be resting on the flat part of the pads, between the center of your fingerprint and the fingertip. I try to make the strings bisect this flat spot when I reach for notes on the neck with my cupped hand opening out of a gentle fist.
As I transition from the neck towards Thumb Position my hand starts to turn towards the bridge, my thumb leaves the base of the neck and tracks diagonally down the side of the base before flipping over into TP. The first three fingers have turned to follow the line of the strings and the index finger is more bent progressively than the 2nd and 3rd. as it stands up over the strings. The strings now make grooves diagonally across the finger ends. All through this change of shape fingers and palm have kept to the idea of a simple bridged "keyboard" shape that hooks the weight of the arm back towards you but allows the bent fingers to rise and fall freely in a flapping motion from the big knuckles (just as they do on the neck). During this transition the left forearm should avoid grounding on the ribs so there is no break in the smooth flow of notes,clarity and tone.
So, now I'm in Thumb Position, with my bridged fingers hooking the string down to the fingerboard and towards my body using the weight of my arm. My forearm and hand are lined up nicely and point diagonally down across the strings. The fingers cannot roll in the same direction as the strings but are free to move widely diagonally. This is OK because only part of the diagonal motion will translate into shortening and lengthening the string (the aim of vibrato).The finger ends will flex and twist because they do not have a lot of lateral strength, and the fleshy pads will also roll a little. Not much movement up and down the string is needed because the notes are now so close together
The diagonal movement of the hand and fingers comes mainly from the flat movement of the elbow, a hinge joint that only moves in one direction. If the fingers and wrist stand up a little (not too flat), and the wrist can flex a little too, I can add a slight roll to the hand motion that will help shape the vibrato and keep it springy and relaxed.
If you go back to sitting at a table with your fingers resting in "keyboard shape" to support the weight of your arm then move your hand sideways your finger ends will flex and roll until they take up slack. There should be a gentle "bounce" at each end as your hand changes direction.Then allow your hand. wrist and forearm to stand up a little and join in by rolling and flexing slightly as your hand moves sideways. Any stiffness in your elbow will affect the the smooth transition from neck to TP, shifting in the upper register and your TP vibrato.
Les Fret, to me the pivot becomes the end of the finger resting on the string and, to a lesser extent (?), the left elbow if the forearm also rolls a little.
Apologies for being so long-winded above but I think that the way the LH transitions in and out of TP and how effectively the fingers can continue to rise and fall strongly, simply and evenly during this process both have a big effect on starting and maintaining vibrato anywhere on the upper fingerboard.
Cheers everyone......
DP