Enjoyment?
There is a web page with an excellent discussion of enjoyment. To make the points in that quickly here, you can make math enjoyable by adding "gum drops", which are enjoyable things that have nothing to do with actually learning math. You can also make math enjoyable by wrapping the math activity around something interesting to the student. You should try to do those with whatever method you are using to teach math. Additionally, the "surfin' math" style also makes math more enjoyable.
Here, the point is that the insight method of teaching is intrinsically enjoyable. Humans were built to enjoy insights, and when the student has an insight of any kind, the student enjoys the insight. So enjoyment and learning occur at exactly the same time -- you don't get one without the other.
So, and here is the point, I watch a student struggle with a problem, then I see the big grin when the student gets the problem. The student is happy. And it makes teaching math a great thrill too.
In contrast, it is basically unenjoyable to recall a method of solving a problem and then apply that method. So the traditional method of teaching math is intrinsically unenjoyable. (If was effective, then we could put up with it being unenjoyable, but it is not effective either).
Caveats
So, the claim is that the insight method of teaching math can be enjoyable. But it only gets this enjoyment when the students have some sort of insight. The insight doesn't have to completely solve the problem, it can just get them closer to the answer. But there is no enjoyment with failure.
You can try to minimize failure. I do. You can offer problems only a little bit harder than the ones the student has already solved. But you cannot avoid failure. The Insight method of teaching math requires setting up a problem for the student to solve, then hoping the student solves the problem. There is no avoiding the fact that the student might not solve the problem.
I don't think that's horrible. I try to minimize failure, but I also think students should learn to deal with failure, and that they will be poorly trained for real life if they cannot deal with failure.
The students all claim that my math enrichment makes their brains hurt. I don't think they see that as a positive. It is unavoidable. Their brain has to stretch to solve the problem, and it is that stretching that builds mental models. So making their brains hurt is a good sign.
The Insight method of teaching math also requires that the student be achieving a goal. Once students are trying to achieve a goal, they can become frustrated when they do not achieve the goal. Again, I try to minimize frustration, but I also believe that good problem-solvers know how to deal with frustration.
Finally, there is an issue of confusion. As far as I know, confusion occurs when facts don't agree with expectations produced by a mental model. If the facts are wrong, there is no learning and the confusion is unfortunate. When the facts are right and the mental model needs to change, the "confusion" is a sign that things are going right. But I have to admit, students do not like confusion, much less welcome it as signalling a potential learning situation.
Overall
My point here is not that the Insight Method is uniformly enjoyable. As noted above, it necessarily has the potential for negative moments, and they will occur. The question is if the good outweighs the bad. Actually, because the Insight Method is effective, we might want to use it even if it was not enjoyable. But in fact students seem to think it is enjoyable.
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