Starting with the reasoning.
All the amplified words are quite short. Usually that would be because the property is very rare, or that they need to be paired with other words of the same length.
In this case, I think it’s both:
An amplified word can have any one of their letters removed, and the remaining letters still spell a word. Without checking the dictionary for rare words, each amplified word has at least three such letters. With a dictionary, almost any letter can be removed.
Here's the complete list of the amplified words, after attacking them with a stack of encyclopaedias:
AID -> id (psychology term), ad, ai (a three-toed sloth)
BYE -> ye (old English), be, by
FOUR -> our, fur, for, fou (Scottish for drunk)
MOAT -> oat, mat, mot (a witty saying), moa (an extinct bird)
PEAR -> ear, par, per, pea
PEATS -> eats, pats, pets, peas, peat
RAIDS -> aids, rids, rads (dosage units), rais (old Portuguese money), raid
SEAT -> eat, sat, set, sea
SPAY -> pay, say, spy, spa
CORED -> ored (OOPS NOT A VALID WORD), cred, co-ed, cord, core
HOOPS -> oops, hops, hops, hoos (OOPS NOT A VALID WORD), hoop
LEAST -> east, last, lest, leat (a water trench), leas (grassland)
PROOF -> roof, poof, prof, prof, proo (used to stop a horse)
ROUST -> oust, rust, rost (obsolete form of roast, valid in Scrabble), rout, rous (OOPS NOT A VALID WORD)
ROUTS -> outs, ruts, rots, rous (OOPS I DID IT AGAIN), rout
SHOOT -> hoot, soot, shot, shot, shoo
The non-augmented ones don’t appear to have this property, except for
SPEAR -> pear, sear, spar, sper (to shut in), spea (genus of toads)
BEATS -> eats, bats, bets, beas (river in India), beat
Maybe I need some particular dictionary to get rid of those two?
EDIT: the dictionary suggested by M Oehm does indeed correctly classify SPEAR and BEATS as non-amplified. Unfortunately, it also does so for a couple of the words listed as amplified. See the edited list in the second spoiler tag for details.