Even without Smoot-Hawley, the Great Depression will still be the Great Depression. The tarriff war it inspired was just sauce for the goose.
Current research indicates that the tariff was contractionary in more ways than simply trade demand itself--It had the effect of constricting credit, and significantly magnified other issues.
But the trade wars were still only part of the story. The incredible and utterly counterproductive imposition of global monetary contraction by the US and French banking authorities strangled the life out of the economy without regard to the broader impacts or the deflationary effects they were exporting through the gold standard system.
Sadly, it's not a simple fix, because the problems in the world's financial system were deeply rooted, and there were political considerations. France for example wanted to accumulate gold reserves in order to defend the value of the franc regardless of the 'rules' of the gold standard, or its effects abroad, and was politically bound by the absolute refusal to tolerate any price inflation or devaluation out of misguided fear for another hyperinflation.
In the US, the Federal Reserve was seized by the notion that the late '20s economy was in a bubble--despite the existence of contractions in some sectors, and the lack of price inflation. They believed they could and should try to pop the bubble by constricting credit, which they did very successfully even as the economy stopped growing. Gold was flowing into the US at the time, which should have been a monetary stimulus to expansion--but the Federal Reserve sterilized that inflow, even while the economy and financial markets were in free-fall.
Trying to pop a bubble with monetary policy is like trying to lance a boil with a mace in the best of times, but the notion that there even was a bubble in the broader economy isn't actually supported by the data. The real economy was experiencing genuine growth in the 1920s, with widespread innovation and diffusion of new products and associated demand for fixed investment joined with growth in productivity. The fundamentals simply don't justify the degree, duration, and overall severity of the contraction until you consider the confluence of
horrible policy moves on a global level. "A" recession was inevitable at some point, but the Great Depression was
not foreordained from on-high, or written in the stars.