I'm relatively sure that it did reduce TP gain when I tested it, but there's a floor() calc done in that process so it'd be easy to miss if you're close to a breakpoint with your weapon delay + whatever Store TP you might have. A quick and easy test would be to just go swing the highest-delay cheap weapon you can equip entirely nekkid (other than the sword strap) a couple times and see for sure. Also, don't just go off of the TP-per-hit values you see on ffxiclopedia - the TP/hit formula was changed a little while after-era, and Nasomi uses the old values (e.g.: with zero STP, a 504 delay g.axe says it returns 137 TP/hit on the wiki, but on Nas it only returns 123 iirc).
If you want to use a TP calculator that still gives you the old-school values, check out:
http://www.evil-kitty.com/tpcalc/ - 'new formula' values will return on top, 'era formula' values will return below.
If the TP return is absolutely unaffected, then either a) the strap actually gives +3% haste and the entire server has been sleeping on it this whole time (or it was recently 'busted' by Nas and no one else noticed yet), or b) it, like The Goggles, does nothing.