A sort of religion has built up around rifle grease, by people who do not understand why it came to be. It was meant to be used in hot-wet environments because it was harder to wash off than oil.
Mil G 46003 Rifle Grease
6. NOTES ; 6.1 Intended use. -The grease covered by this specification is intended for use in the lubrication of rifles and other small arms only when such weapons are used under conditions of sustained rain. The temperature range under which this grease will be -used is 2OC (35F) to 38C (100 F) . This grease is not intended as a replacement for conventional oils under conditions other than sustained rain.
The physical characteristics of GI Rifle grease were spec'd out to be a NLGI type 3 grease. Basically the Army tested greases in hot, wet environments, lubriplate 130 worked, and they used its physical characteristics in future procurements. Without calling for lubriplate by name. You can find pictures of guys dressed up in raincoats holding a rifle. A huge hose pumping warm water was sprayed on the shooter, the rifle was fired until it seized up. What a miserable job!
Still, had to be better than the Helmet testing job!
My opinion, modern grease technology is better than what was around in WW2 and the lubrication requirements of the M14 are not that severe. If you like grease, than a good waterproof grease, like the tubes of outboard motor grease found in big box stores will work, Lubriplate 130 or equivalent, or use the more expensive stuff. They are all good, just keep the thing lubricated. I use grease in hot weather and LSA or motor oil in cold weather. In extreme cold, the rifle is to be wiped dry of all lubricants.
