The form lhegin in the published Etymologies might be a misreading for lhegrin, see VT/45:25. As noted by Bertrand Bellet, the two forms are listed side by side, and they may simply be doublets, but it is also possible that we have here a singular followed by its plural.
Primitive elvish
alak
root. rushing, rushing, [ᴹ√] swift
rindi
adjective. swift
A root whose most notable derivatives are Q. alqua, S. alph “swan”. The earliest iteration of this root was ᴱ√ḶKḶ from the Qenya Lexicon of the 1910s (QL/30); the other derivatives of this root from this period have to do with “appearance” such as ᴱQ. ilk- “to seem” (QL/42). By The Etymologies of the 1930s this root took on its later form, ᴹ√ALAK and had the gloss “rushing” with other derivatives like ᴹQ. alako “rush, rushing flight, wild wind”, N. alag “rushing, impetuous” and N. alagos “storm (of wind)” (Ety/ÁLAK). It was also an element in the name of S./N. Ancalagon “Biting Storm”. Given the continued appearance of this name of The Silmarillion (S/252), the 1930s meaning of this root may have survived, but it is hard to be certain since the name was only properly translated in the 1930s.
The 1930s root also had an unaugmented variant ᴹ√LAK with derivatives ᴹQ. (a)larka, N. lhagr “swift, rapid” (Ety/LAK²). Whether this unaugmented variant remained valid is unclear, but there is nothing in Tolkien’s later writing contradicting it either.