Absolutely. And as you say, those people do not consider themselves transgender. My point is that transgendered individuals do qualify gender as a huge and fundamental part of their existence, to the point that they are willing to make serious changes in their life in order to address it. It's a spectrum, not an on/off switch.
I'm also a little concerned that you seem to be regarding transgender issues as some sort of personality defect based on the latter part of your post. If modern attitudes and studies have taught us anything about sexual and gender preferences, it's that they're not some blown fuse or crossed wire that can be ironed out like a bad habit.
Whether transgendered individuals become what they are via nature or nurture is besides the point - development is development, and "undoing" these kinds of deep-rooted aspects of personality isn't something that can realistically (
or ethically) be accomplished.
As mentioned, it varies. There are many people though who are content to live with the genitals they were born with however, whether it's because they want children or simply because they're able to live their life as the opposite gender without getting hung up about what's under their clothes. If for example a transgendered individual is in a long term relationship with someone and both of them are content with their bodies, there's really no reason to subject yourself to that kind of serious surgery. Who else is really going to be seeing what's down there and making an issue of it anyway?