On what basis do professionals who recognize gender dysphoria support risky and permanent medical intervention for those seeking gender re-assignment?
I hope I am qualified to answer this question. I am not a psychologist but have worked with transgender persons in volunteering positions a number of times and am fairly well researched on the topic.

The biggest key point as I have studied simply is based on health outcomes as opposed to comparing this to mental illness.

This page talks about the practical health outcomes of the treatments. Individuals who identify as transgender already often suffer from a variety of issues that reduce both their quality of life and in some cases health. These can include developing mental disorders if they do not get treatment and even self harm or suicide. Overall most of the linked studies in that article showed that the benefits largely outweighed the risks in an overall health decision.

Now you imply that other treatments are used for most conditions which have effects on mental health, usually either "talking cures" or medications. The difference being in almost all cases these treatments are ineffective or even make the situation worse in transgender individuals. 

I cannot find a study to link to this which is publicly available right now but in short, treating transgender individuals with treatments used for mental illness usually makes the condition worse. In many cases it causes additional depressive disorders to form as well as long term self image problems and self hatred, as well as increasing the risks of self harm or suicide.

So in summary, it is not to say the treatments used for transgender persons are risk free, but merely that they are better than the alternatives both of doing nothing and of trying to use a purely psychiatric method to treat. As both of those tend to make the condition gradually worse and no other treatments are known which produce a consistent improvement.