To be perfectly honest, I have trouble remembering what the potential confusion is here for people (though obviously there is one for certain folks), because the intent in any given situation involving melee weapon attacks or melee-weapon attacks in the rulebooks have been clear in my experience.
If they only use the phrase one way, where's the confusion?
The confusion comes from whether "melee weapon attack" means "an attack with a melee weapon" or "a melee attack with a weapon" throughout the text.
In technical/formal English, the lack of a hyphen signifies that each adjective independently modifies the noun, whereas the presence of a hyphen forms a compound adjective that itself modifies the noun. So, technically speaking, in the phrase "melee weapon attack", "melee" and "weapon" both modify "attack", resulting in the phrase meaning "a melee attack with a weapon".
If instead a hyphen had been included, the compound adjective "melee-weapon" would modify "attack", so the phrase "melee-weapon attack" would mean "an attack with a melee weapon".
The problem is that the hyphenation rules aren't widely know and/or followed, so in casual English the hyphen is often dropped. Unless a reader knows that the author both is aware of the rules for compound adjectives
and that they followed them correctly, it's impossible to know whether the lack of a hyphen in "melee weapon attack" is meaningful. Those with editing, formal writing, or technical writing backgrounds are likely to spot the potential ambiguity (in commonly do so, in my personal experience). Best practice when a compound adjective is not intended (i.e. when the formal rules say not to include a hyphen) is (arguably) to reformulate the sentence to avoid potential confusion. The designers did not do such a reformulation here, so the meaning of the phrase in the rules does indeed hinge (and deliberately so, according to the SAC quote) on the lack of a hyphen.
The ambiguity matters in the rules for any ability that requires a "melee weapon attack". Interpreted as a correct application of the compound-adjective rules, any such ability works with any melee attack made with a weapon (or without a weapon, actually, thanks to the unarmed strike debacle). Interpreted instead as an
incorrect application of the compound adjective rules, any such ability works with any attack made with a melee weapon, regardless of whether that attack is melee or ranged. Ergo, the ability would also work with thrown weapons.
As it turns out, we know that the designers used the compound-adjective rules correctly, so it's confirmed that abilities requiring a "melee weapon attack" do not work when throwing thrown weapons, even though such weapons (except for darts) are melee weapons. But that still leaves us with a text whose correct interpretation depends on a single hyphen and a somewhat-obscure rule of writing that readers have no reason to be confident that the writers followed.