There is little logical reasoning to take the test. You can do it at home yourself.
Sex Determination: The SRY gene (Sex-determining Region Y) on the Y chromosome triggers the development of testes in an embryo. Without this gene, a fetus will develop female reproductive organs.
So, if I look down, I can see that clearly a Y chromosome was present during my fetal development.
But there are other conditions that can also affect an intersex condition. It is possible to have both male and female organs. That is something a Y-chromosome won't tell you.
Then there is AIS (Androgen Insensitivity Syndrome) where the body has the Y-chromosome, testes are present but the body is essentially immune to the effects of testosterone. This results in a body with feminine shape and appearance, except for those dreaded dangly bits.
There are many other possibilities, and Kleinfelter's is only one of them. For most of us, the only thing the test will tell you is if you have multiple X's along with your Y. That really doesn't answer most of the questions we have.