whaaa? What if the inbound was late due to “ATC” or weather?

In my case, the inbound was 15 minutes late. But does that really explain why our flight was delayed for four+ hours?

This is why I said "(usually)", because there are exceptions.

It's not such a clear cut, assuming this goes to court the airline will likely have to prove it has taken all reasonable measures to avoid the subsequent delay of the passenger stemming from the late arrival of an aircarft.
When we're talking on a flight departing from the carrier's hub that might be difficult to prove, but when we are talking about a long-haul flight from Singapore delayed for only a few hours it's more likely the airline could prove that.

In your case, since this is non-EU to non-EU flight which doesn't fall under EC261 delay compensation, why would it matter?

insurance

I don't know why they are trying to hide something that is so obvious. Delta doesn't hesitate to admit when they messed up (in my very limited experiences dealing with them under similar circumstances).
https://cimg4.ibsrv.net/gimg/www.fly...c8c8f8c98b.jpg

I don't know what do you mean by hiding, you can get exactly the same on the KLM website - https://www.klm.com/flight-status?da...irportCode=AMS
You can even go back and check the previous flights the aircraft did.

Of course it could be that the details there are incorrect/inaccurate, as you said the incoming was delayed but it might not properly account for the total delay on departure.

The certificates that are downloadable from the flight status page make no mention of the reason for the delay. It only confirms the delay, which we already know, so those certificates are useless. Oddly, my other flight shows up on the website as delayed due to technical issue (iirc they told us it was a catering issue), but it won't allow you the option to download the certificate from the website to confirm the delay.

Would a screenshot showing the reason not suffice? Alternatively the insurance company can check themselves on the KLM website.
It might be my ignorance but I don't see how a simple PDF which is not signed or restricted in any way is better than a screenshot or a link to the website.