I have had no personal dealings with Kumon. I have some friends who had their children in it for a while, and I know several people who did it as children, and I know one person who taught at a Kumon center for a while. They varied in their opinions of it.
My understanding of the Kumon method is that it focuses overwhelmingly on computation, and that the method's strongest proponents are unapologetic about this. I do not mean to suggest that they should be apologetic--- I just mean, they are entirely aware of this focus and they do not see it as a liability. Frankly it would surprise me to hear someone suggesting the Kumon system as a way of promoting any kind of generalized "understanding" of math. Whenever I have heard people talk about the Kumon method, it is promoted as, in decreasing order of frequency,
(a) a way of raising flagging grades in mathematics, due to a student's difficulty with getting the right answers, or taking too long to do so,
(b) a supplement to a school curriculum that is seen as lacking in computational essentials,
(c) a way of making a student's calculational skills "automatic", so their mind is free to focus on the non-calculational aspects of their non-Kumon math education.
I have never heard of anyone using Kumon to directly promote "conceptual understanding" of mathematics, and if that is what you want to invest in, my personal opinion is that Kumon is not the way to go. (I don't really know what would work at that age, beyond giving your son access to puzzles, books, and people who are enthusiastic about mathematics.)
Additional random thoughts based on my own experience as a tutor and teacher:
I would not evaluate any method of instruction of 5 year old children by the speed with which it acquaints students with the fact that order does not matter in addition. Many concepts like this--- concepts that generalize or abstract large numbers of individual facts, each of which can be verified through a computation--- take a long time for people to internalize. (For one thing, the language one really needs to express general relations like this is the language of algebra, and it is not taught until much later.) I do not mean to say that students at age 5 cannot grasp that order doesn't matter, only that it takes a long time for this to really "sink in". People forget about things like that, and needlessly duplicate calculations, even at the university level.
I no longer teach math, but when I did, I found that the longer I taught it, the less sympathetic I was to the idea that it is harmful for students to see math as a bunch of rules to follow. It is harmful for them to see math that way if (i) they are never given a coherent set of rules, or (ii) they aren't competently taught about how to use the rules, or (iii) they never personally practice what it means to follow symbolic rules, or (iv) they never develop any idea of where the rules came from. Your own example (that $x + y = y + x$ for any two integers $x$ and $y$) is itself a rule, and as you argued in your own question, it is immensely helpful to be able to recognize instances of this rule. From this point of view, your objection to Kumon (in that instance) is only that they haven't gotten to that rule yet.
Having said that, I don't know don't know how well the Kumon method really rates by the standards (i), (ii), (iii), (iv) just listed. My guess is that Kumon makes very little attempt at addressing the last point, and this is a legitimate source of concern, if you expect Kumon to fill that role. The person I knew who taught at a Kumon center said they were extremely restricted in what they were supposed to teach--- she didn't go so far as to say there was a script, like telemarketers have, but almost sounded like that. I think this would preclude any meaningful discussion of the "real meaning" of a lot of calculation. At the same time, I do not think your son will be getting that in his regular math classroom either. It is extremely difficult to teach basic mathematics "conceptually" and most elementary teachers are not competently trained to do it.
Long story short, I think Kumon is more about (a), (b), and (c) above than it is about "conceptual understanding." I personally probably would not use Kumon except for reasons (a) and (b) above.