Do you remember what He-Man actually did with his sword in the cartoon? He didn't fight with it—at least not swinging it against his enemies. He mostly used it for things like hacking through chains and deflecting the beams of magical energy his enemies shot in his direction. Here is a typical example, using his sword to turn aside a dragon's fire.
When he had to fight physically against an evil minion, He-Man typically resorted to fisticuffs—sometimes fighting one-handed because he still had his sword out.
The reason He-Man never used his sword as a melee weapon was that many makers of children's programming (especially in the 1980s) were trying very hard not to show any realistic uses of weapons that children might want to imitate. Filmation, who made the original He-Man and the Masters of the Universe cartoon series, were particularly careful about this. Yet they still took a great deal of criticism for making the show too violent, even though, as the head of Filmation points out in this 1985 New York Times article outlining the controversy,
"We try not to have He-Man hurt any living creature, and the good guys always win," said Lou Scheimer, president of Filmation, the animation studio in Reseda, Calif., that produces "He-Man" and "She-Ra."
This is the same reason that Hank the Ranger from Dungeons & Dragons (1983) never seemed to use his energy bow to shoot at any enemies. And it is why Superman (1988) took place in a near future setting, in which villains used ray guns rather than realistic handguns.
The absence of Skeletor's matching sword was another manifestation of this. If Skeletor had still had his matching sword, there would logically have been reason to have the protagonist and villain face off in a duel. (I remember plenty of boys playing with the original He-Man and Skeletor action figures, the two hacking away at each other.)
In fact, practically none of He-Man's enemies had weapons (at least not that they used in battle). When melee was unavoidable, He-Man would win using boxing and wrestling moves. There were no visible wounds, and nobody seemed to be permanently injured.