Simple answer is NO....as we are not a Nanny State or People's Republic of....yet.
Until the Hawgs are considered "Game" under the Hunting Laws...no licenses required.
No licenses required for the guns.....and probably not any time soon either. If one is shooting an "Automatic (read machine-gun) then a Federal License is required (400 USD per weapon as I recall).
The FAA under Part 135 might have some purview but if the activity is not "commercial" then not then either....beyond the usual Part 61 and 91 rules.
Remember what personal freedom was all about? Ours is being chipped away at bit by bit particularly under the current administration but we still have quite a bit yet.
Despite all those endorsements, training, protocols, regulations, laws, tariffs, and procedures.....you still have mustering accidents....and make things complicated and expensive as all Hell. I reckon we should stick to our method....simple...cheap...effective.
You are using one video to pass judgement on everyone that is doing the Hawg Shooting.....perhaps others do it a bit differently...and maybe even those in the video have changed their methods...as no one intentionally goes out to get hurt. If they do....then no matter how many rules, tests, etc....you require...they are going to succeed in doing so.