There's no surer way to get the Dems in Washington to suddenly support border control than have the migrants shifting the Overton window to the right.

I don't think so. Mass immigration advantaging the Dems electorally has not moved Republicans so I don't see why the reverse would be any different.

the leftists get their way because the status quo keeps the border weak

Republicans are also getting their way.

As it is now, every day that the politicians are "tied," we lose

Republicans have controlled the federal government for stretches since mass immigration became a glaring problem. They are part of the problem.