Ok, several points there so I'll take them 1 at a time.
People do sometimes have it about their bodies - for example people who have limbs removed because they feel "wrong".
Body dysmorphia is definitely a thing - the limb one, anorexia etc. However, the vast majority of trans people (as currently defined) don't have it. And in all other cases of dysmorphia the treatment is not to affirm it, but to offer therapy to help people accept their bodies as they are. Surgical removal of limbs for body dysmorphia is banned almost all over the world, and nobody would dream of treating anorexia with gastric surgery.
I don't think people want or need to make sense of it as such.
The average person on the street, for most social purposes, probably doesn't need to. But legislators and regulators do - they can't just wave it away.
Regarding the penis, one could argue that the penis does not maketh the man. A man could lose a penis in a terrible accident and still be a man. Ergo, if penis does not equal man, then penis does not NOT equal woman.
That's a Mad Hatter argument. It looks as if it might mean something at first glance, but not when you actually examine it. If a man who has his penis remains a man (which I agree with), then a man who still has his penis is also very definitely a man.
Now competitive sports, this is a very tricky one IMO but I have seen various studies or medical information / Dr quotes etc that suggest in some cases there is no natural obvious inherent advantage for over sex over another etc.
That is true for a small number of sports - many of which are already not segregated by sex (such as showjumping). But for any sports in which there is a male advantage, that advantage is retained by transwomen. Hormone treatment may - over sufficient time - reduce muscle mass (although still not to women's levels) but it doesn't shrink men's skeletons or lungs, it doesn't change their pelvic angles or skull thickness .... etc. It does not make competition with women fair.
There are also examples of eg intersex sportspeople who've been categorised one way or another and subsequently had a tough time.
Not sure what exactly you mean by 'a tough time', but DSDs are medical conditions entirely unrelated to trans identity.
Finally, there's also the argument it's harmful for trans people to be excluded from sports as it "others" them.
They are not excluded from sports. Transmen can comepete in their sex category (provided - like other women - they don't break doping rules); transwomen can compete in their sex category. Everyone is included!
What they are excluded from is competing in the category for the opposite sex. And it is considerably more harmful for women to be deprived of places, rankings, prize money, scholarships, and to be physically injured in contact sports, than it is for transwomen to feel 'othered'.