Regarding the "Trump is an idiot", I have a different thought. Maybe he is very smart in using the leverage now, harvesting the fruit planted by Obama? Using the refugee leverage to get something more out of Australia? See, he is not (yet) blowing things up, but ONLY threatening, that's exactly how one does. Maybe soon Australia will concede something, and we will see Trump fulfill his "intention" of honoring that agreement?
The agreement still gives Washington leverage over Australia, but less than if the arrangement had not been called into question
publicly by Trump.
The issue is important for the Australian government because it has made a public commitment that no asylum seekers arriving by boat will ever be resettled in Australia. This message is intended specifically to deter asylum seekers from coming to Australia, and is the culmination of a generation of policies pursued by Australian governments designed to satisfy domestic political constituencies demanding "secure borders". The former Prime Minister John Howard put it succinctly when he said that "we will decide who comes to this country, and the circumstances under which they come."
It is never admitted as such, but the disproportionate political focus in Australia on "boat people" basically acts as a safety valve for the expression of broader societal anxieties about immigration in general. Something like 30% of Australians were born overseas, a higher ratio than any other country of significant size, higher than USA and much higher than UK. At the same time, the level of anxiety about immigration in Australia, although growing for similar reasons as in other western nations of late, seems lower than most of those nations. And unfortunately, I think the fact that successive governments have managed to redirect the bulk of anxieties toward a small subset of prospective immigrants, i.e. boat people, is one of the reasons that there has not been a stronger backlash against immigration levels more broadly.
So the Australian government is caught between the political imperative fulfil its promise that no asylum seekers arriving by boat will ever be resettled in Australia, and the lesser problems that are posed by indefinite detention of asylum seekers in other localities, some of whom have been there for years. The direct costs of such policies are high, the arrangements with foreign government precarious, the criticism of conditions from the UN, humanitarian organisations, and the political left (a category in which I include myself) in Australia unrelenting, and there are regular "incidents" that bring the matter to national political attention and controversy, such as those in which asylum seekers commit suicide, including by immolating themselves, or a pregnant asylum seeker in a country that does not permit abortions, or hostile relations between current and former asylum seekers and local peoples, or reports into the mental wellbeing of, especially, children who are in detention.
The deal between the Turnbull government and the Obama administration therefore promises to take a great load off the Australian government by all but emptying our current offshore detention facilities of asylum seekers, some of whom have been there for years. And therein lies the leverage that the agreement gives Washington over Australia.
But by
publicly calling the arrangement into question, Trump has already caused the Australian government a significant degree of embarrassment. Trump is by no means popular here, and his objections have highlighted the fact that the Australian government is beholden to his whims. In this light, if the deal were to fall apart, Turnbull would be able to say -- or rather project indirectly -- that, look, we did the best we could, but the new US administration is just unreasonable and does not honour its word. It's possible that Turnbull could even spin the development to his political advantage by projecting that he "stood up to" Trump on behalf of Australia and did not cave to his unreasonable demands. After this public dispute, the domestic political costs to Turnbull of failing to implement the agreement are much lower than they would've been otherwise.
On the other hand, if Washington had sought to use its leverage privately, e.g. "You don't want B-1 bombers based in Darwin? Wouldn't it be unfortunate if something were to happen to the asylum seeker transfer agreement you signed last year", that could well have been an effective source of political pressure. Once tensions are aired in public, matters of national pride and the domestic political standing of politicians enter the picture, especially when the figure on the other side is as widely disliked in Australia as Donald Trump.