Breaverman is definitely not an honest actor in all this. She has no intention of curbing immigration or fixing the refugee crisis. It is, as you said, just inflamatory rhetoric designed to rile up the tory voter base.
However, the left and mainstream political discourse is also firmly entrenched in a strong pro-immigration stance, and often only because it's seen to be the right thing and it pisses off the "gammon". For instance, they are unwilling to accept that France is doing to us what Turkey is doing to Greece, and what Greece does to the rest of Europe. If they manage to get in, move them onward toward their destination asap. There are "charities" literally ferrying these people across the med and english channel.
We've had Songatte and Songatte 2, then the jungle. It's not getting better, and it never will because the push factors driving these people into Europe will only get worse. Even if we streamline the refugee intake process, it won't make a difference, the demand is essentially infinite. If the barrier to entry is reduced then more people will start popping up in Calais.