The first clue that the video would be waste of time was the text below it. Take note: I've never once saw someone call people that disagree with them 'snowflakes' and then heard something useful dribble of their mouth. This one was no exception.
A knife is a hunk of steel, no more and no less. Some knives are more suitable for batoning and some are less so, but certainly some of the best knife makers I've ever heard of feel a quality knife should stand up batoning just fine. I'm not even sure why it's controversial. If you don't want to baton with yours, don't do it. But to insist no one else should smacks of a thick blend of arrogance and stupidity.
Knives are made by humans, and things made by humans can break. Will batoning wood with a knife break it? Maybe. Sometimes it breaks axes. A guy I know posted pics of this at another forum a couple days ago. It was a pretty nice Council axe and this guy has a nice collection of axes. If it was made by Man it can break.
I am probably messing up this quote but gist of it is that axes make legs shorter and hatches make arms shorter. That's pretty spot on. It's much safer to split small pieces of wood with a knife than with an axe IMO, and if you use the latter you should consider laying the piece down long ways. For splitting large rounds certainly a maul is better, or a wedge and a hammer. Better still, a hydraulic splitter if your pack is big enough.
Hikermor has often said he rarely needs to cut or split wood. Fair enough, in a lot of cases you don't need to. Certainly a small cooking fire is easily managed in many locations with just what you can pick up off the ground. That depends on a lot of things, of course. In many cases if you want an all-night fire you may have to cut wood (although of course, maybe you won't).
I very often baton wood for kindling. In the area where I live there's a lot of wood but there isn't much for twigs like in some kinds of forests. In the fall you will often have dried pine cones but eventually I usually need to cut at least some wood and sometimes split it. For splitting down to kindling I haven't found anything as easy and accurate as a knife. Certainly it's faster! In the clownish video at the top, I'd have been done cooking and have my stuff packed up by the time the fella had his little wedge made!
TL;DR- Hike your own hike. Life's to short to rant about how terrible it is that other people do things differently than you do. Use an axe, use a knife, use karate like Sponge Bob does!
Just don't even thing in your dream at night that your way is the only way.