Good point. I agree with you concerning their motivation.
But, as usuall, they greatly misappraise the importance or significance of "the fight."
It's two bald men fighting over a comb.
The template of the left is that conservatives are stupid. This will reaffirm that stereotype. This fight will provide great entertainment value for them, and will bind them together to continue their other efforts against conservatives, because in their minds nothing conservatives do or think is valid.
This fight is important to them, and they will win it.
Big Time.