Tooth to tail is crappy PC/corporate-approved rename. The concept used to have a bunch of arrow and spear related names and a bunch of informal phallic counterparts all of which are better suited to the fundamental workflow of mechanized offensive operations.

For curiosity’s sake, what were these things referred to historically and informally?

The spear analogy dates at least as far back as 1940. Heavy mechanized units as the tip and then reinforcing units behind them, further units to secure flanks, logistics flow. From the metal differences across the head of a spear (something that people who have a passing familiarity of artillery or naval gunnery will be able to relate to) the finish of the handle to keep the wood in good shape the analogies basically write themselves. The dick jokes are an obvious follow on.

Reading between the lines, it sounds like it was probably one of "tip"/"point"/"head" and "shaft", or similar

That will certainly resonate with the generals (≧▽≦)

If the tail is gone, the dentures become useless