I am well aware of the war on general computation, but I fail to see how it's relevant here. If you are saying you don't want to be alive in a world where this war has been lost, that's... a rather strong statement.

To make an analogy, we're slowly losing the ability to fix, modify, and, ultimately, control our own cars. I think that is highly unfortunate, but I'm unlikely to declare a full boycott of cars and go back to horses and buggy whips.

Since you're basically talking about security, you might find it useful to start by specifying a threat model.

how trustworthy such NNs have been at performing their intended functions properly and at avoiding being subverted

What do you mean by "such NNs"? Neural nets are basically general-purpose models and your question is similar to asking how trustworthy computers have been at performing their intended functions properly -- it's too general for a meaningful answer.

In any case, the point is that the preference for open-source relies on it being useful, that is, the ability to gain helpful information from examining the code, and the ability to modify it to change its behaviour. You can examine a sufficiently complex trained NN all you want, but the information you'll gain from this examination is very limited and your ability to modify it is practically non-existent. It is effectively a black box even if you can peer at all the individual components and their interconnects.