Can we "prove" that slow growth is due to deficient demand?
This is one of the issues with Macroeconomics - you usually cannot prove these kind of things, you can only find indicators.

That is, because what we observe is always - in the neoclassical language - the general equilibrium, never the "off equilibrium adjustment" (as in the Walrasian Auctioneer). That includes Keynesian logic - these days, the majority of Economists does not think about these traps as off-equilibrium paths, but rather as multiple equilibria [citation needed].

As a comparison, if you only observe the equilibrium outcome, it is hard to say whether a decrease in prices comes from an increase in supply or a decrease in demand. Now, if you could randomize or had a natural experiment at hand (say, randomly shut down half of the firms in the economy), you could actually "prove" the underlying issue with empirical data.

But since this is not feasible within many subjects of macroeconomic interest, disproving ideas using empirical evidence is quite difficult.