It’s true I was assigned a sex at birth. So were you. The difference is my gender identity did not match that sex. That is the actual definition of transgender: the sex assigned at birth and the gender identity do not match. Whether I transitioned or not I would still be transgender. MANY trans folk cannot afford to or otherwise choose not to medically transition. Many trans people also are in a position where it isn’t safe to socially transition. They may be financially dependent on parents who would kick them out if they came out as trans, for instance. And legal transition-name change, gender change on IDs, etc — can also be costly and might depend on a medical transition (even changing gender marker on passports requires a doctor letter), so many people don’t transition legally for some of the same reasons as medical or social transition.
As someone who has transitioned, I can say that yes, it was a process, but it was not the defining process you make it out to be. I would still be a woman whether I transitioned or not. The only difference is you and others might refuse to recognize me as such.