My experience from proving a moderately complicated result in my PhD was that it's neither. There wasn't enough time to brute force by trying many complete solutions, but it also wasn't a single flash of insight. It was more a case of following a path towards the solution based on intuition and then trying a few different approaches when getting stuck to keep making progress. Sometimes that involves backtracking when you realize you took a wrong path.

Yeah agreed - there are actually many, smaller flashes of insight, but most of them don't lead to anything. I once joked that you could probably compress all the time I was actually going in the right direction in my PhD down to about a month or two. That's a bit glib, often seeing why an approach fails gives you a much better idea of what a proof 'has to look like' or 'has to be able to overcome'. But many months of my PhD were working on complete dead-ends, and I certainly had a few very dark days because of that. Research math takes a lot of perseverance.