This narrative framing of everything being the GOP's fault stops, right now.

No one's framing "everything being the GOP's fault." The GOP and the Democrats have mutually failed to adopt any significant immigration reform to address their differences since 1996 (the 1996 bill was mostly a GOP wish-list on immigration, which Clinton pushed as part of his efforts to triangulate for re-election). The two parties have different positions on the border on a wide number of issues.

There's "nothing stopping the Dems" from doing what Republicans want on the border, just like there's "nothing" stopping the Republicans from adopting universal single-payer health care - other than the fact that a wide swatch of the voters in their party don't support those measures. The Democrats want to ensure that people who are legally entitled to asylum in the United States continue to have a path to claim it, so that we don't repeat the horrors of WWII (when we turned away a boat full of Jews fleeing the Nazis). They're willing to compromise with the GOP on a number of border security measures in order to address valid issues at the borders - but it is inconsistent with that core value that the MS St. Louis scenario could be repeated.

It takes two to compromise. This narrative framing of the GOP being entitled to 100% of what they want on border security or else it's the Democrats who are being unreasonable stops, right now.