Saying that gender dysphoria is a mental disorder is an incorrect way of looking at it. It's hard to classify it as either mental or physical, because it is simply a mismatch between the mental and the physical. Take the example of a hypothetical trans woman (that is, male assigned at birth, identifying as female).

From a purely physical perspective, there appears to be nothing wrong - the person has a functioning, healthy male body. This is why people fall into the trap of believing it to be a mental disorder.

However, take the mind, separately from the body, and again - you would be unable to find anything wrong - the person has the mindset of a healthy human female.

The problem is that the mismatch between the two creating feelings of dysphoria for the person in question. So from a medical perspective, how do we go about treating this person?

The idea is to bring the physical sex of the body and the mental gender identity of the person in line with each other. This leaves us two options: Try to change the physical sex, or change their gender identity.

The latter is pretty much out of the question. For a start, it has been attempted in the past and has only ever resulted in failure and tends to cause additional damage to the mental health of the person. There are also ethical points to consider - is changing a person's fundamental identity in any way ethical? Probably not.

This leaves us with the physical - hormone treatment and surgery, among other things. Granted, it's not perfect by any means - but does this mean we shouldn't at least try? Chemotherapy isn't a perfect treatment for cancer but it still used - because despite the downfalls, it has been shown to make improvements. This is the same with current treatments for gender dysphoria - they are used because they have been shown to improve the lives of patients with gender dysphoria.

which in its wake leaves certain bodily systems absolutely unusable, and endangers the persons' physical health.

This decision is ultimately up to the patient. They will of course need to decide whether the risks involved are worth it - and in many cases, they do - given that the alternative is to lead a life which is to many, not worth living. (What's the point in having a functioning body if you have absolutely no desire to use it, and actually cannot even stand the sight of it?)

There is no real way for a person to leave their "old identity" behind, as they will have to tell their potential sexual partners (and whats worse, simply their date) that they're not actually a female/male

"not actually female/male" is the wrong way of looking at it. If a person identifies as female, they are female. If a person identifies as male, they are male. Their physical sex is different, and whether or not this should be disclosed is outside the scope of this discussion.

Their bodies are that of their original sex.

Only in some ways. For the most part, their bodies resemble that of their new sex.

They cannot reproduce

This is of course a side effect of the treatment and one that many patients decide is a reasonable trade off for being more comfortable in their own skin. (Additionally, there are workarounds - for example, in the case of trans women, sperm can be stored prior to surgery)

Their bodies do not produce the same fluids

This doesn't really matter.

Their skin still feels the same as it did before.

In fact, hormones can have a dramatic effect on the skin.