@Nebulous :
As others have said, Magic Weapon cannot be cast on a monk according to RAW.
That said,
IGNORE IT!
As you surmise, it is a pretty stupid lingual verbiage IMO. Giving a character a +1 to attack and damage rolls for up to 1 hour at the cost of a spell slot
and concentration (enough of a handicap in 5E, also IMO) will not hurt a single thing.
Allowing a paladin to use an unarmed strike and channel his divine strike through it increases the flavor and magical feeling in a game.
Casting other "weapon only" spells on unarmed strikes also doesn't hurt a thing and just makes the game better IMO.
Also, although Unarmed Strike is not
currently in the list of weapons in the PHB, it WAS before:
Removing it, and making it a "melee weapon attack" which is not a melee weapon was DUMB and has led to so much confusion and debate it is moronic. The rules worked simply enough before, allowing melee attack features to apply to weapons, unarmed strikes, natural weapons, etc.
Numerous tweets and Sage Advice has come up on this
because they made it needlessly confusing.
Although not RAW, my advice to
anyone playing D&D is forget JC and the other designers' nonsense. Like many things, the had it right the first time and later changed their minds, adding special this and special that (concerning what is an unarmed strike, etc.) to cover it all.