The errata says:
“Instead of using a weapon to make a melee weapon attack, you can use an un-armed 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.”
It's intent is clear. "Instead of X you can do Y" is the same as "You can do Y instead of X".
Reversing the sentence, it reads "You can use an un-armed strike to make a melee weapon attack instead of using a weapon".
Thus unarmed strikes count as melee weapon attacks for the purposes of the OP's question. Or would do if the DM wasn't being strangely obstinate.
The issue arises because of the phrase "melee weapon attack" but it exists to differentiate between melee attacks from spells ("melee spell attacks") and from not-spells (hence "melee weapon attack").
I'm not sure what other term would cover it, tbh. "Melee physical attack", maybe? Though that sounds strange.
Simpler to acknowledge that, while you can use your fists (or head, elbow, knee, foot) as a weapon (by using them to physically attack someone), it doesn't make them an actual weapon.