There is nothing in the text of the amendment which requires any criminal conviction, civil settlement, judicial review. If you're interested in a really detailed breakdown and argument for why Trump (and many others) are disqualified from holding office, I highly recommend reading The Sweep and Force of Section Three, which was the paper that really set all of this 14A S3 discussion in motion. It is very long, and very detailed, but it's a relatively easy read, in mostly plain language. I haven't even gotten all the way through it yet, I'm still working on it. Much of my understanding of this issue comes from this paper, but believe me when I say I'm not just waving it aloft and declaring that it's gospel. What it says makes sense.
With regard to your question about each state officials' discretion to exercise, there's another comment in this mess of a thread where I go into some more detail on that. I also made note somewhere that, because this is constitutional law, state officials who refuse or neglect to follow the Constitution may be in a position where they could be held personally responsible for failing to apply the law if they don't disqualify people who are disqualified by 14A S3.
Edit: For the record, Rosenberg stated early in that podcast that he favored Ford pardoning Nixon. That pardon played a huge part in the situation we find ourselves in today. It set a precedent that presidents can commit crimes without consequences. We had an opportunity to set a fair and equal precedent, and we did not. Rosenberg being okay with kicking that can down the road doesn't give me a great deal of confidence in his opinions.