Why is gender reassignment surgery advocated to be “required” for health insurance purposes, but “not required” for legal purposes?
The term I think you mean to use is “Medically Necessary”. Medically necessary procedures and medicines should be covered by medical insurance. Whether a medicine or procedure is medically necessary is a question best left to the patient and their doctors.
Not all transgender people need to have surgery in order to effectively deal with their gender dysphoria/incongruence, but for those who do, it is a necessary part of their treatment.
As to why it isn’t a requirement for legal recognition, it’s for a similar reason – not all transgender people need surgery as part of their transition, but for those who need to be legally recognized as the gender they experience, surgical status should not be a consideration.
Why is it impossible to change your gender identity?
Let’s make sure we’re on the same page. Gender identity is the personal sense of one’s own gender (see here). So for a gender identity to change, a person’s sense of their own gender must change.
The way the question is written, it asks “why is it impossible to change your gender identity?” (emphasis mine). When you use the verb “to change” in that form, it is an active action – the person is taking action to enact the change. So for it to be impossible, that would mean that nobody has any capability to act to change their own sense of their own gender. I’m not convinced that nobody has that ability. People who are genderfluid/genderflux experience their sense of their gender as changing, so obviously the sense can change.
But can someone affect the change? While I’m not aware of anyone who can make their own sense of their gender change, I’m not willing to make a blanket statement that nobody can.
So I’m going to conclude that the premise of the question is flawed. It is possible for one to change their own gender — we just might not yet be aware of how or how often it happens.
Gender World 
Gender identity cannot be changed no matter how hard you try. Gender identity is between the ears and not between the legs. There is no magic blue pill to cure us of our gender identity, all medical can offer us today is a medical transition.
Some transwomen need the surgery and other transwomen are comfortable with their penis. From a legal standpoint, would you like someone dictating to you on whether you keep or lose your penis. This would be a recipe for disaster.
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