Apologies for not gauging the difficulty well when I asked the question. question_asker was very, very close with their answer.
The answer is
The solfege scale (do, re, mi, fa, sol, la, ti), which describes a major scale in music.
I am the thing most needed in all the world,
or "kneaded", rather: Do pronounced like "dough"
but so fast nothing can catch me.
Re, pronounced "Ray", as in ray of light, faster than everything else
I myself can go the distance
Mi, pronounced like "Me", a name for myself. And Fa, pronounced like "Far" with a british R, referring to distance.
even until the nearest star.
The nearest star to Earth is its sun, named Sol
Articles have been written about me, though not one in English.
The part-of-speech definite article La used in several non-english languages, such as Spanish.
I give a soothing ending to an evening's rest.
Ti, pronounced like tea, which is sometimes an evening drink, though it's a bit of a stretch
But be warned, I stand in judgment of all things measurable,
Reference to scales in general? Musical scales in a more general sense dictate a lot of rules of our music.
and this day, I could be at your throat.
Singing sounds are produced in the larynx, which is either in or near the throat.
The main character in the Sound of Music teaches the children to sing using solfege in the song that starts "Do, a deer, a female deer" and this may be the most well-known instance of a song featuring/teaching solfege.
28.6% = 2/7. Light and light's maker are Re ("Ray") and Sol respectively, which make up 2 of the 7 syllables.
Another song in the Sound of Music is "The hills are alive with the sound of music". So music is that which gives life to the hills, and is a stepping stone to the more specific answer.