Maybe the canon has changed over the years, but my recollection is that the person wielding the UN had to have a strong will or risk being destroyed, too.
And I'm not saying that Doom doesn't at times show mental fortitude. I'm saying as of the writing of this story he was clearly not at his peak. Quite the opposite. Even he himself admits to his counterpart just before killing him that he was mentally exhausted. I found it odd that in that state he would use the Nullifier, instead of, say, the Cosmic Cube, which was also in the display case and is powerful enough to accomplish the same end with zero risk to Doom.