The Dems are short on leverage because about half the Dems, politicians and the general population, are border hawks and want less immigration generally. Even in a presentation designed to show how much different Dems are:
56% of dems want to fine you for hiring an undocumented migrant
20% of dems want to ban travel from certain Muslim-majority countries
16% want to build THE WALL
28% want to end CHAIN MIGRATION
11% want "Separating families who cross the border illegally as a deterrent"
https://www.npr.org/2018/07/16/62884...-trumps-agenda
If you look for polls where people obviously wanted worse seeming answers (it's in how the questions are phrased) you'll find worse numbers for democrats.