Fossil fuels and automation have made us wealthier for the past century and a half. The overall pie has grown, but labour's share of the pie has been shrinking for the past few decades.

It's why in 2018, you can be living paycheck-to-paycheck on minimal wage, but own a flat-screen color TV. Is this a better life then doing the same job for more pay in 1950, where there were no flat-screen color TVs, home appliances were much more expensive, and much less useful? In many respects, yes.

The question is, what will your life be like in 2050? If current trends continue, we expect your share of the productive output of society to go down. Will this be offset by productivity gains (More, cheaper stuff)? Or will limits to growth (Cost of energy, cost of pollution) actually make you worse off?

It's possible that automation worked to decrease labor share in that period, but other factors(communism, GI bill, etc...) had a stronger influence in the other direction.

Also, general economic growth due to underlying fundamentals in the demographics -- fundamentals that western societies don't enjoy today.

Sure, that’s possible. But unless we can point to supporting data, it just amounts to seeking to reinforce our preconceived notions.

Of course it's not possible to show cause and effect between labor share and abstract 'automation'. Especially since not all automation is equal in this regard. Automation in some areas may increase labor share (accessible automation like PCs and washing machines) while others decrease it (expensive/in-accessible automation like assembly lines and machine learning). This means that automation could increase labor share one decade and decrease it the next as the types of automation change.

However, the argument for a current/recent causative link between the two is not a "preconceived notion" but rather a reasoned hypothesis arrived at based on observation of trends and application of logic to economic theories.

It is more reasonable to suppose that:

overall, automation has had a role reducing the labor share (which has been going down despite a couple of brief spikes in the 50s and 60s)

than to suppose:

overall, automation has been increasing the labor share, but this is overshadowed by other causes which have been decreasing labor share.

Thus, if you want to argue for the later supposition, the burden of proof is more heavily on your shoulders to explain the other causes which somehow outweigh the effects of automation on the economy.