I would, and have, recommended a .410 for HD. Why not? Yes, it has fewer pellets but how much does that matter. Four pellets of OOO buck shot at 1200 fps vs. 10 at the same velocity.
It matters by about 60% less wound channel. Air in, blood out, structural incapacitation, or overpressure shock are the 4 general ways projectiles stop attacks. Shotguns and handguns only get to use the first three. Buckshot is marginal at the third, so air in and blood out means more, or bigger holes, not less.
Buckshot pellets, are in essence, small caliber ball. There is little to no stretch cavity and they get pushed off path rather easily. The benefit of #1 buck over 00 buck comes in the form of more pellets and thus more tissue damage. The stopping potential of buckshot is only through more paths, not less. So, yeah it matters a good deal.
In those cases I have worked on professionally as a ballistics expert there are some departures and similarities between handgun and shotgun stopping power. I will assert that in most cases, it does not matter....22LR, 9mm, 00 Buck, Shotgun Slug, .223, .308...as far as stopping the threat. Sure, the death rate increases as you move up, but more times than not, a single shot stops the attack. It is those cases where it does not, where the minutiae of "stopping" power gets argued.
There are a LOT more cases of shotguns used criminally available for forensic analysis than cases used by the good guy. But in that realm, the survivability of those shot with shotguns (bird to buckshot, but not including slugs) is higher than those shot with normal defense/duty caliber pistols. It is not parallel since the victims are not homicidally motivated, but the direct mortality rate could be transferred to a probable difference favoring the handgun for stopping an attack. I am sure there are successful and unsuccessful stops with .410 buck, but the only two uses I have looked at were woefully unsuccessful.