Why did many socialists and organized labor across Europe support the First World War...

Why did many socialists and organized labor across Europe support the First World War? Why did they choose the nation over international solidarity? It amazes me.

Because the instability gave revolutions an opening, and that's exactly what happened in Russia and nearly happened in Germany.

Because they thought socialism would gradually happen rather than outright revolution
Bolsheviks were the minority of socialists

*hires freikorps to annex bavaria*

Why did they support a war where workers would kill each other protecting the status quo? They weren't neutral, they actively supported this.

Because SocDems didn't want to lose seats in the next election

More or less, they defaulted to the popular issue, instead of sticking to their principles. Simply put, anyone who did not support the war was sidelined initially. Besides, war propaganda has effectively framed the war as patriotic defence, instead of petty struggles of the overlords.

Nonetheless, the public opinion did change drastically as the war dragged on, and the politicians with anti-war convictions could capitalize in the end.

Instability caused by devastation and strain of war, mind you.

There were different justifications for it. For example the German Social Democrats argued that Russia was the most repressive, backwards, anti-worker country in Europe, so if Germany could impose a new regime in it then it would help the Russian working class. On the Entente side many argued that Britain and France, being the cradles of the French Revolution and modern democracy respectively, were more progressive than the Central Powers, especially the Austro-Hungarians and Ottomans. In reality I think that these were just justifications for many socialists succumbing to nationalist sentiment, mixed with people who genuinely believed that their side winning would be an immediate benefit for the socialist movement. Like said, most of them at the time would be considered a form of reformist or at most peaceful revolutionaries today. People like the Bolsheviks and Spartacists were firmly in the minority among the movement, so many of them thought that actual revolution was a far off prospect and that more modest gains were more worthy of their efforts.

The issue is that they did betray their initial policies that existed before WWI windup.

Like I mentioned , this was an organic process caused by public opinion of the general populace.