The hypocrisy of it is the entire point. It's a demonstration of power. Combined with the rhetoric what it clearly communicates is "play the farce or we'll do this to you too."
They don't suck at PR, they are doing exact what they want and it is extremely effective. You simply are coming into it with the wrong frame, one based on viewing it in the detached and abstract hypothetical, vs the embodied view of someone who is in that society and has everything to lose.
Everyone ultimately knows the Big Lie is false. The PR is showing that the truth does not matter vs the power of the regime.
One of the most chilling examples of this is the footage from when Saddam took absolute power. He paraded out a "confessor" that showed signs of torture in front of the legislature. The confessor started reading a list of names of sitting members of the legislature. As he continued, the entire congress became panic'd. Many of them began screaming hyperbolic pledges of loyalty to Saddam, prostrating in the aisles, etc.
Once the list of names was done, roughly half the legislature was named. Saddam took everyone outside, distributed firearms, and made the half that was not named execute the half that was. He did that to make the survivors complicit in his new dictatorship. The message was: if I go down, you go with me now.
That's what this sort of stuff is about. Not some armchair platonic musing about whether it's hypocritical vs the ideals of democracy.