The words are in the right order, they just left out the comma after “better”. See my post 15 and all will be clear.
But, I’m really just endorsing your idea from a slightly different path.
It’s a matter which is the cause and which is the effect. I was worried that someone would call me on the grammatical error in my version (use of wrong tense). I’d have put in a disclaimer; but “brevity is the sole of wit” and all that. (Or is that soul?)