You’re confusing transgenderism with transsexuality. This man can have a vagina and be he/him, it s really his business if he wants to take testosterone and grow a beard whilst having a vajayjay. Meanwhile, a MtF woman can have a dick if she did not go through sex reassignement surgery. In this instance, both are not transsexual. The trans man is female (vagina=female) whilst socially being a man. The woman has a penis (male sex organ). This confuses some people and is why it is still a debate.

The real reason this all makes social practical sense is because a lot of males in society are not very manly and a lot of women are not very feminine. In the past the words to describe these people would be faggot, queer, dyke, or other slurs. Simply because they did not fit what society defined as “a real man” or “a real woman”.

It’s also important to highlight that a lot of transsexual men and women do not believe in transgenderism because it leads to non-binary genders, genders which are unclear or vague (like the "they/them which means the person feels equally masculine and feminine). To these trans men and women, sex reassignment is acceptable but since they do not understand the nuance between gender and sex (or they do not accept it) like a lot of cisgender people who lean right politically, the debate keeps occurring because of willful ignorance whilst this debate has been closed for over 20 years within academia.

Correct, males and females are sex designating terms. Men and women are the byproduct of social uses of biological terms (the sexes) which resulted in social characteristics we wound up calling genders. Recently we have added genders following historians discovering that other cultures, aside from our roman/pagan one, used more than genders than the binary ones. This being an obscured fact due to the colonial age, during which European settlers destroyed the cultural patrimony of the majority of other societies on this earth (often now referred to as the third world or global south). And before someone makes the mistake of calling these ideas woke, these ideas are the product of decades of work from historians, sociologists, archeologists, anthropologists and other doctors of the social sciences and are contested due to a gap in education between the average working class human and the average academic.