Yes, I understand the pump process for these lasers. Don't forget that Nd:YAG is also used instead of ortho vanadate, though the Hades probably is Nd:YVO4.
A bare diode is highly divergent and the beam specs aren't great (unless we're talking about VCSELs). The DPSS process though gives a chance to correct for this. Remember, this isn't some magic crystal we stick in front of a diode to change its color. The process is far more intensive than that in ways I don't even really understand to the fullest, but the general characterists of the DPSS process and lasing within the crystal just allow better beam specs. Keep in mind though, with the bare diode you're limited by the properties of it to poor beam specs. However when you pump vanadate, the vanadate becomes the laser. It's not some random optic along the way, it re-radiates a whole new laser, with a whole new set of (better) beam specs.
If you don't believe me, feel free to check up the specs on these lasers themselves. The manufacturer specs on the bare diode 808 and 980s are <5mRad and 6mm at aperture, whereas the 1064 Hades matches the 532nm Aries at <1.5mRad and 1.5mm at aperture. That's a HUGE difference, and you can see it in the lasers. The bare diodes ones diverge horribly if you want a small diameter at aperture. If you want good divergence, you get a huge initial diameter.
I don't doubt that, using a weaker lens, you can get great divergence, but your initial dot size will also be pretty large. You can see this on the Glowtorch, WL pulsar, etc. You get somewhat better divergence than the greenies, and in exchange a HUGE dot size. If you check out specs on bare diode and DPSS lasers across the board, you will see the DPSS are generally FAR better than the bare diodes ones.