It makes it more honest, for one thing.
And second, it may sound as a paradox, but those authoritarian leaders are more connected with the masses and their needs (and vice-versa) compared to a pseudo-democratic and very opaque assembly composed of some hundreds of members (what are now called “parliaments”) of which no-one knows all that much, for that matter.
So a switch to an authoritarian leader would in effect, and comparatively speaking, mean a return of effective power to the masses, because it is on those masses that the legitimacy of said leader depends. Today’s parliamentarian regimes have long lost their legitimacy (notice senile 90+ years old ladies being parliament members in the most “democratic” country on Earth), it’s all a Glasperlenspiel [1] at this point, following prior moves blindly while the reason behind those moves has long been lost.
[1] https://en.wikipedia.org/wiki/The_Glass_Bead_Game