It's hard to assess economic stability because big crises are low-probability events, but it doesn't seem obvious to me that Bretton Woods US (01944–01971) was systematically less stable than the US after the switch to fiat money in 01971.
Certainly it was far more economically prosperous; the change to a fiat currency coincides with a widely-remarked drop in the GDP growth rate and an even more dramatic growth of inequality, to the point that the lowest quintile in the US has famously seen no real income growth since the switch to fiat money.
But that could easily be a coincidence, because many other things also changed. For example, the Bretton Woods period was a historically anomalous period during which the US was by far the world's dominant economic power; all the other advanced economies had been bombed to smithereens, the PRC and Russia were additionally crippled by communism, and the former colonies in India, Africa, and Latin America were only beginning to recover from colonialism. The dollar itself enjoyed the exorbitant privilege of being the backing for all other currencies. Plausibly the US's dominance during this period allowed it to strip resources from weaker powers in a way that dramatically enhanced its own economic growth, a way that is no longer possible.
Also, you could reasonably quibble that, rather than being a natural division between systems, comparing to Bretton Woods amounts to historical cherry-picking: it's been 51 years since the US transitioned to a fiat currency, and the preceding 51 years include the crash of 01929 and the Great Depression! Any period of world history is economically stable compared to a period embracing the Great Depression.