Linking economic and political developments is always going to rest on claims about causality, and those will only be as good as the underlying causal models. There isn't really any way to strictly distinguish whether Hitler came to power because of the hyperinflation in the early 1920s, or the deflation of the late 1920s, or both, or neither, because there is only one history of Germany. If we want to reason about what might have happened if, we need to dive into counterfactual reasoning. (Not that I object! Just that this is necessary to understand the question.)

The hyperinflation of the early 1920s has generally been exaggerated in "folk" economic history as a direct cause of the rise of the Nazis. Mostly, this is a matter of sheer chronology: the hyperinflation was ended around the same time as the Beer Hall Putsch, in late 1923. The hyperinflation almost certainly contributed to the erosion of trust in the Weimar government and its ability to solve the country's economic woes. The recent memory of hyperinflation may also have contributed to the later willingness by Bruning to endorse full adherence to the gold standard, and the associated austerity. But, these contextual matters aside, it is hard to see a clear chain of causation from hyperinflation to the rise of Hitler. It was decisively ended by the mid-1920s, and eclipsed by rather different severe economic problems in the early 1930s. One might say that hyperinflation was one of the causes of the Beer Hall Putsch, as a symptom of a weak state and a cause of unrest. But the Nazis were a fringe paramilitary organisation at this point, and not a major political force. They were in no position to seize power, and the result of their coup attempt at this early stage was failure and imprisonment.

The argument that the Bruning austerity led to the rise of the Nazis, by contrast, is quite clear, and well-supported in the literature on the Great Depression and the interwar economy: the attempt to balance budgets, pay reparations, and still remain on the international gold standard led to austerity in the form of tightening government budgets and contractionary monetary policy; austerity led to mass unemployment and wage cuts; depression led to unrest; unrest led to the fear of violent left-wing revolution, which in turn led to the unholy alliance between the moderate right and the far right; this alliance then provided legitimacy and influence for Hitler and the Nazi party, leading to their takeover and the end of Weimar democracy. None of these effects need be sufficient or necessary, but the argument works so long as these effects made the Nazi takeover more likely.

The economic links in the chain are clear and not widely contested. The logic of the gold standard is usually understood in modern economics in terms of the macroeconomic trilemma: the impossibility of having free capital flows, fixed exchange rates, and independent monetary policy simultaneously. The international gold standard means abandoning independent monetary policy: in order to maintain the exchange rate parity, painful procyclical monetary policies needed to be implemented, which in turn meant the government had to either forego deficit spending, or find willing buyers for its bonds. So long as investors in the US were wiling to lend, the latter was an option, but after 1929, there was a sudden stop and reversal of capital flows. Thus, the only remaining options were austerity, or to abandon the gold standard. Pro-cyclical austerity causes unemployment and income loss, and the Great Depression was the most severe trough in the business cycle in history up to that point.

This, then, was the immediate economic context for the Nazis seizing power: a destabilised and unpopular government, attempting to make good on its international commitments (including reparations) and to maintain the fixed exchange rate of the gold standard, at the cost of widespread unemployment. Unlike in the case of hyperinflation, this deflationary recession was contemporaneous with the increasing popularity of the Nazis, their initial electoral successes, and their eventual seizure of power. This is a much more plausible case for an economic cause of the rise of Hitler.

But we must be careful not to exaggerate. Germany was under tremendous economic and political strains during the interwar period, and it was not alone in this. The entire world financial system was fragile, overextended, and by 1929, collapsing. Austerity was one poor option out of many other, equally problematic options, each carrying severe risks in terms of alienating political groups, inciting violence, destroying international cooperation, collapsing economic output, and so on. Austerity policies in the late 1920s very likely did make a significant contribution to the rise of the Nazis, but it is not so easy to know what else could have been done, and in any case, history can only ever be an imperfect guide to the "what if" questions.