Hi Morri, For sure too long a cable that binds in speedo head will cause needle bounce. How much does it bounce at what speeds?
Look at top end of cable & see where it seats on speedo. With cable hooked up at drive end measure how much cable is sticking up above the seat on cable housing. Record measurement. Now take a skinny flat blade screw driver & stick it up inside speedo head. Record that. Look at end of speedo. Do the math & see if your cable is too long into speedo head. Don't just grind as a guess.
Next feel how hard or easy the speedo turns using a flat blade screw driver. It should feel very free with only slight resistance.
Now hook cable to speedo head & remove it from the drive. Turn cable. It should still be quite free. The inner wire should turn with only slight resistance when screwed into speedo head.
It's unlikely the drive is faulty. In any case pump grease into speedo drive. Like 5 strokes of grease gun. Any excess will slowly ooze out over 100 miles or so. Will not cause any damage.
The cable must be well lubed. eBay
sellers have dedicated cable lube which works good. However you can use wheel bearing grease smeared on wire, then a smear of motor oil to dilute & soften the grease. I know you're not supposed to lube upper 6-10" of cable, but I've found out of desperation the top bend just below speedo head can generate a lot of friction. I know your not suppose to lube the speedo bushing, but I've found 1-2 drops of 3 in 1 oil can work wonders.
Pretty much most vintage Triumph speedo needles wiggle to a degree, but only about 2-3 mph or less. Wild fluctuations are almost always friction winding up wire, then friction is suddenly overcome & cable unwinds suddenly causing needle to jump several miles per hour. First if falls as cable winds up, then raises as cable springs back. This oscillation gets magnified by momentum of the needle & magnetic cup.
Magnetic speedometers have a spinning magnet at the top of the shaft. There is an aluminum cup surrounding magnet. The cup is attached to bottom of needle shaft. As the magnet is rotated it generates a tiny electrical field in cup. The electrical field is magnetic. So this magnetic field is now being rotated around the spinning magnet pulling needle around. There is a small clock spring shaped return spring on needle shaft. Delicate stuff. So any oscillation of the spinning magnet must be removed.
My hunch is when spinning with drill it's smooth so needle is fine. But connecting cable you are getting binding at some point. So you need to experiment to find out where the binding is occurring. You are going to have to find the binding point.
The suggestion of running bike with rear wheel off ground is good. When doing this disconnect cable from speedo. Lightly grip the spinning cable end with finger tips & see if you can feel any oscillations of cable rotation.
One user found the cable was dropping too low & the square end could skip in speedo head. He ground down a nut as a spacer & slipped it over cable to raise cable end up. The cable wire can be pulled out.
If needed you can grind/file a spacer washer & put it on top of cable housing to move housing away from head or drive.
There is a cause for this, I know you have it inside you to find it. No question about that.