@exotic
"I don't like it because when I have taught DCs who do it I find them inflexible-they tend to state 'that is not how we do it at Kumon' and they don't want to even consider another method."
It has been a few years so my recollection is a bit vague but I think it was to do with column addition. Kumon taught DS the 'adult' way i.e. you add up the columns from the right and put the carry over on the column to the left. DS got into trouble with the teacher because she was teaching the method whereby you add the columns from left to right and she pulled up DS for using the adult/kumon method.
At that stage he had been doing column addition for a few months in Kumon where he was adding up multiple 4 to 5 column figures and here was this teacher telling him that he was doing it wrong when he could see that the method she was using only worked for small numbers. DS ended being quite upset because the teacher was not receptive to his 'that is not how we do it at Kumon' observation.
We spoke to the teacher after school one day and she accepted that it only works when a limited number of columns is involved but it is be the recognised method of introducing the concept of column addition to young children.
What wound me up was that faced with a kid that could use the adult method she still insisted that he use the kids method.
I obviously can't comment on your scenario. All I am saying is that my son was faced with a situation where the kumon method was the correct method.
"I consider that a good mathematician is the one who makes connections, knows why something is done, can pick the way that suits them and can apply it in other areas"
I agree with you there but, on the positive side, Kumon did equip DS with a Human Calculator mind :) so when it came to do 'proper' maths he could do the calculations in his head. And as I've said above, kumon is just an aid and not a total solution even though Kumon HQ might market it as such.
"To be fair, the children who go to Kumon are not the good mathematicians and they like a crutch so they can say 'this is....... and this is how you do it-but that is too rigid an approach"
This is the point I made above. Some parents look for help because their child is struggling with maths. Kumon comes along and says this is how you do it, follow these steps and you will get your answer. Given the above, of course you are going to end up with a child that is rigid in his approach and doesn't really have an indepth understanding of the subject.
"The really good mathematicians are more likely to play competitive chess etc."
That is only true in our culture. The common view here, and probably in the Real World, is that the OP's child is doing fine so there is no need for external learning whether it is Kumon or something else. So of course the kids who are already good at maths won't be signed up to do Kumon and Friends.
In my case, DS was good at maths and we put him into Kumon because we wanted him to be even better. Such a view isn't popular in our culture which is why a significant proportion of the Kumon kids that we saw at our centre were Asian and Oriental.