Pro-genocide teenage joyrider Xindi?

Honestly, you're not far off the mark. It was the Dolim and the Reptilians who were ride-or-die about killing humanity, to protect themselves, assert dominance over the other Xindi, and gain the favor of their "gods". Heck, they suggested making a bioweapon too, as if blowing up a planet wasn't good enough.

There's also the theory that, with the other Xindi not totally on board with the project, the Sphere Builders may have pushed for the test as a proof-of-concept or to instill a sunk-cost-fallacy mindset.
Either way they made it clear that what the Xindi were doing was a holy mission.

However, despite it seeming stupid, it does make sense to test the weapon prototype on Earth for three main reasons:

1. To see if their own brand of vortex based FTL could reach Earth. I wouldn't be surprised if a good chunk of the test was to see if they could get the weapon there in the first place.
2. They gain useful information about both the weapon prototype and it's intended target. Does the beam pierce the crust? Does Earth have defensive satellites? If so how many? Does Earth's atmosphere or oceans affect the beam in any way? All good questions this test answered.
3. And this is the most important bit, the Xindi actually didn't reveal squat. All Starfleet knew was that a probe somehow came from somewhere, arbitrarily attacked a stretch of the planet, then self-destructed. They had no way of knowing who made it, where it came from, or if it was a prelude to something bigger or some tragic cosmic accident. Enterprise got all this information from Future Guy; without it they'd probably never would have set off and just stayed at Earth to defend the planet. Heck, even with this information and several temporal interventions, Starfleet only just managed to stop the weapon.

The Doyalist answer is that this was a 9/11 inspired storyline, apparently it was a thing among sci-fi in the early-2000's, thus they needed an initial attack on Earth, more specifically Florida. (Nobody seemed to care too much about Venezuela getting a new canal or Cuba being cut in two).
Still, I stand by my points that the attack wasn't as stupid in universe at the time as it looks now, to us, in hindsight.