Is it true that buying the STI in bad times and selling it in good times is far more profitable than buying the S&P 500 over the past 25 years? How do you prove this?

This is a good, but only a theoretical question because these are different investing strategies,and how about comparing against buying the SP500 in bad times and selling it in good times?

Evidence shows that even professional fund managers are not capable to market time,

even less so then we retail investors. How can one have a definition of 'bad times' and 'good times' as entry/exit points? This would be the basis to answer your question for data.

There could be a hint in the following longterm chart (STI versus SP500), that suggests STI had more volatility, but then remains the question: how could one have known, the optium entry/exit points (my feeling: completely impossible, but only post hoc)

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From the chart, ait was clear that SPX gave better return becuase of high volatility. But last Sunday Invest section mentioned by Bernard Tan "If one had "traded" the STI from 2003 to 2007 and again from 2008 to 2019. The total return would have ben 690%. These returns are astoonishing as Singapore, which is not a market often associated with high returns , beat the mighty S&P 500"... Any one can do a back test or simulation to prove the case.