Lots of commenters on the video wondering why they didn't make the blade sharper, since the axe they made was so dull it might as well be a bludgeon. Woulda helped if they explained that the crystalline structure and brittleness of ice makes it really hard to sharpen an edge. Though they did mention that a sharp edge would be rapidly melting back to bluntness, so that's something.

Out of curiosity, I tried to see if anyone else had attempted an ice weapon before. Judging by this video, ordinary ice is absolutely hopeless. Polyester-reinforced ice seemed to have promise, but even with extensive sharpening they couldn't make a knife sharp and/or sturdy enough to puncture a flimsy water bottle.

https://youtu.be/r1GzOadFYpg

If the axe material could take it, you could probably still decapitate someone by whacking their neck off with enough brute force. Probably need a guy a lot bigger than the one they used in the demo to pull that off, though.