@Saeviomagy
I can't see how you'd read it to be any other way than the OP stated.
I explained how, so you have indeed seen it. You may not agree with it (as I'm sure you don't) but it's there

With that said, just remember my original comment...
That is how I read it currently in my highly distracted state. My mind could be changed for sure if I saw something more I might be missing, but at the moment I don't see it.
-- that's how I saw it last night during my first take
@Gruns
Umm.. Of course it does. At least as far as anything that cares about whether or not your are attacking with a club is concerned... Otherwise, what is the point of saying it? If Belt of the Brawler wanted you to simply deal 1d6 damage with +2 proficiency with unarmed attacks, they'd have said that. I'm not sure I understand why you think it wouldn't apply here...
Hmmm... yea, I don't know... I guess I was looking at it in the most simple way...
It says "When you use those weapons" (club or mace) - and we're talking about using Spiked Gauntlets (not a club or mace). The Belt does not say anything (to me) that would lead me to believe it is over-riding that simple statement. It says "as if you were using" which to me says "you aren't using".
They may have worded it that way so that it would be clear (yea right) that your fists were still pounding out damage as opposed to, say, slicing or piercing... or perhaps it was so that it was clear that it was not magical damage... or perhaps so that it would be clear the damage could still be increased by a damage category in the case of use by a Bugbear/Minotaur, etc. Who knows.
I don't know, I'm not WoTC and I'm just throwing out thoughts/impressions as they come to me as I mentioned above and as I also stated, a ruling for my games has not been set in stone in this case and in fact I
would allow it as the OP posted (even if it seemed to me it was not the intention).
I was explaining it to a friend last night that as far as damage output it didn't really matter to me as there are other ways (obviously) to achieve the same damage output - I don't see (yet) anything overpowered, etc that would keep this off my table.
With that said, I can see your point, why would they not then just say it did extra damage.