But being unable to disengage from the Big Problems and live your little ordinary life is not heroism, and it actively gets in the way of solving any of those Big Problems.

Not my Big Problems; they get solved from doing just that.

How do you know? The question isn't whether obsessing fixes the problem; it's whether taking breaks speeds up the overall process. You don't need tons of hours to fix the problem; as you said earlier, a few minutes to explain the right insight is quite sufficient. What you actually need is the right few minutes of work, spent finding the right key insights.

Thinking longer about a problem is only helpful to the degree it produces new insights. As you've found, this can be very inefficient. If taking a break and not worrying about an unsolved problem increases the efficiency of future problem-solving even a little bit, it could well be worth it.