It’s a bit of a “hack”. It cools down the polyfuses, which trigger if they’re too hot.
It probably cool down the triacs a little, which make them less likely to blow up.
I don’t think it’s a necessity, but wouldn’t remove mine since it’s a 2-3$ part (from China). You can replace your fan easily with a 80mm computer fan. Either find an old PC in a dumpster or get a new one from a computer store (shouldn’t be much more than 10$).
I think you’ve changed your power supply output to 13v if I’m not mistaken? That might be why the fan got really hot. You may have better luck with a higher quality fan. But then again, 13v vs 12v isn’t much of a issue:.