The original grease was Delco ST800, a clear grease that worked from -20 to 100 degrees. You need to clean off the old stuff & use a similar type grease. I use a refigeration type clear grease that falls within the above specs.
My Corvette(s): 1966 white roadster, "Plain Vanilla"
So if the odometer drive gear isn't plastic, what are all those little white plastic looking chips that crumbled off it where the final metal gear drives the odometer tumblers? The odometer stopped rotating when the teeth fell off.
The original speedo head came with steel gears. Someone has replaced yours with a service part and cheaply put in the nylon gears designed for 68 up cars. The midyear needs steel gears to drive all the other gears. Nylon gears strip faily quickly when put to the test, as you have found out.