...the club and his trail arm are about in line at the start of the video, then the club shaft and head seem to take off ahead of the torso.
That is true. The angle between the club and his trail forearm is opening much faster than his body is turning.
Also at the beginning of this video, is Hogan making a big effort to swing his arms, or are they being puLled by his torso? Are his hands and grip completely relaxed at this time.
In this clip I don't see any arm movement independent of torso movement which is what I'd expect with 'swinging' arms. I doubt if he's significantly changed the force of his grip but to get to a potential point you may be asking about, no I don't think the action is caused or even aided by forceful straightening of his trail wrist or a pulling of his lead arm independent of his shoulder turn. If you try this yourself (when your foot allows) you will see that neither is necessary with the illustrated trail elbow position.
Looks like with todays science they could put sensors on
different parts of the body and see what does what in a golf
They can and they do however many in the golf world don't care for the results. For example there is a study that shows that the maximum pressure applied by the trail hand on the shaft for expert golfers corresponds to exactly the point you see in this clip and that the pressure then goes to near zero at impact. Not a suprise when you see this clip with the trail forearm acting as a fulcrum at this point but with less effect as the angle between club and trail forearm increases.