That post only works if you use an absolutely ridiculous definition of hate speech.
It does not show a flaw in the opposing argument. It's not an example of a rule being applied arbitrarily. It's an example of falsely claiming a rule applies when it objectively does not apply. If that's a weakness, it's a weakness in basically every law. It's not a disqualifier.
They could have tried to show an unfairness or a contradiction that actually relates to sloppy definitions of hate speech, but they didn't. They went outside the definition.
> instead of applying them arbitrarily (which you won't like when your political party isn't in power), just apply them fairly across the board regardless of whose in charge
I think you mixed something up here. Applying a law fairly doesn't prevent a different political party from interpreting it differently and ending the fairness. I'm pretty sure your argument is supposed to be that these laws should not exist in the first place, because they're too dangerous in the wrong hands. Not "just apply them fairly".