I would say it depends on what the "training" is for. Most of your montage seems to be a display in the air, which there is no point in having a sharp axe for, a full weight axe maybe, but not sharp. Chopping wood, absolutely, it needs to be sharp and I've read and heard many times of the conditional benefits from chopping wood.
This must be one of the most ridiculous things I've ever seen on Youtube. Sledgehammer drills are a real thing, but an axe? Also, the music in that video was so annoying I shut it off after ten seconds. I skimmed through the narration a little but as far as "martial arts", no thanks. Defending yourself with an axe is probably going to end poorly for you. Not to mention, it's not a particularly agile tool, and I expect the person in the video will eventually sever an artery. These exercises are safer and probably far superior as far as workout quality.
The big difference here is that Mayweather is just doing this for strength training. It's practically the same drill as with the sledgehammer posted above. He's not swinging it in made up patterns pretending to be a Viking like the guy in the first video. And the more protestors watch that first video the more silly it seems. Would you approach this guy in the park? Not me...I'd probably call the cops
I would absolutely call the cops if I saw someone swinging a real axe around in a park. Also, if I saw someone throw an axe towards a target with no backstop and the flying axe went too high and sailed beyond its target. !!! (0:50). Of for that matter, anyone throwing a real axe in a park.
