How did the silver standard "insulate" China from the beginning of the Great Depression (1929-1931)
As argued by Ho (2014) the silver standard helped China because downward trending silver prices were producing mild inflation during the period 1929-31 and also helped to devalue exchange rate between gold and silver.
First, as already discussed in this Economics.SE answer, mild inflation is important for an economy to avoid excessive unemployment due to nominal rigidity of wages especially in the recession. In fact this is one of the many reasons why the conventional wisdom of combating recession is expansionary monetary policy (see Blanchard et al. Macroeconomics a European Perspective).
Furthermore, it is worth noting that Great Depression was so severe in US because US monetary policy went in exactly  opposite direction to the above mentioned conventional wisdom. As famously argued by Friedman and Schwartz in the monetary history of  the United States the Great Depression would arguably be only mild recession if Fed would not decide to pursue disastrous contractionary monetary policy.
Second, as discussed by you and also in the article you linked increasing gold prices and decreasing silver prices allowed the exchange rate depreciate vis-à-vis gold standard currencies   and currency depreciation is another conventional way of countries fighting recessions (again see Blanchard et al mentioned above). Normally, this policy can be risky because other countries could also competitively devalue their currency leading to potential currency wars, but in this case as their currency depreciated gold standard currencies appreciated.
Also, I disagree with your assessment here:

But after looking at historical prices of both metals leading up to and following the Great Depression, I saw divergence, but not dramatic divergence. So my intuition has been depleted.

According to data from macrotrends the price of gold rose between Jan1929-Dec1931 from approximately ${\\\$}315.38$ to ${\\\$}402.71$ (growth rate of $\approx +27\%$), while  price of silver decreased from ${\\\$}8.07$ to ${\\\$}5.17$ (change of $\approx -35\%$) in the same period. Or in another terms the ratio of prices $Gold/Silver$ changed from approximately $39$ to $78$ (approximately doubled) - that is quite a large change in exchange rate. I would be surprised if such appreciation of Gold vis-à-vis Silver would not result in some non-trivial boost for Chinese exports.