It is actually a weapon attack (a melee weapon attack, to be precise), but it isn't an attack with a (melee) weapon. An unarmed strike cannot be used for Divine Smite RAW, because it calls for "a weapon" in the text of Divine Smite. However, you can use it for the Hasted action "as it says "one weapon attack only".
The official sage advice disagrees with your reading. That is fine, but it is something you should accept.
And as the sage advice is how the rules are "supposed" to be read, that means that something in the divine smite rules makes unarmed strikes not qualify.
Instead of using a weapon to make a melee weapon Attack, you can use an Unarmed Strike: a punch, kick, head--butt, or similar forceful blow (none of which count as weapons). On a hit, an Unarmed Strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.
This can also be read
Instead of using a weapon to make a melee weapon Attack, you can use an Unarmed Strike: a punch, kick, head--butt, or similar forceful blow (none of which count as weapons). On a hit, an Unarmed Strike deals bludgeoning damage equal to 1 + your Strength modifier. You are proficient with your unarmed strikes.
Here "unarmed strike" replaces everything before "you", not just "a weapon".
That reading is then consistent with "you cannot smite with an unarmed strike".
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All English is ambiguous. Hell, all formal logic is ambiguous (if less so).
Rules text is English, especially in 5e. So you have to accept there isn't one single valid reading of rules text.
Instead, you can find multiple consistent readings.
In this case, under the SA ruling that divine smite is not doable with an unarmed strike, to maintain consistency you may have to stop thinking of unarmed attacks as a melee weapon attack.
Prior to that SA ruling, the idea that "unarmed strike" was a kind of "melee weapon attack" was a pretty dominant reading of 5e rules. But that way of reading it does not look consistent with the "unarmed strike" doesn't work with "divine smite". But adjusting the way we read the 5e rules can remove "unarmed strike" is a kind of "melee weapon attack" while leaving most of the core gameplay unchanged; instead, you can use an "unarmed strike" in place of a "melee weapon attack" during the attack action (much like you can substitute a grapple). And the game keeps on working, and you cannot smite with an unarmed strike, and Paladins can't punch harder than monks.