Others seem to have mutilation covered, so I will tackle a different angle.
There seems to be an impression in the world that SRS is a casual thing that people wake up one day and do. That someone decides 'Huh, I want to change my genitals!' and that's that. The reality is far, far removed from this.
As it is, there is a huge humilation factor in the process. Very, very few surgeons with take you without notes from psychologists and doctors, evidence of passing the so-called 'real life test', in which you are expected to live 'full time' as a woman, and personal interviews with them or their staff.
Getting to the point of being 'allowed' SRS is measured in years. I was a fairly quick example; I had my SRS two and a half years after deciding to transition, and about a good 20 years after I realised I was 'wrong'.
I have known for as long as I can remember that my body is wrong. As young as age 5, I was a girl in my dreams. I lived in a relatively conservative family, and learned early on to hide my difference, but it was there.
It was, for me, /very/ much a physical thing. I've never particularly wanted to wear 'women's' clothing (though I do love the freedom to do so!), or wear pink, or other 'socially feminine' things. But I also knew I wasn't supposed to have the anatomy I did.
In my early teens I found the internet, and outlets for my growing unhappiness. Role playing games where I could be a girl, people who understood, information on it all.
Things went from there, but that's not really relevant. I didn't decide until I was 28 to actually risk transitioning. And yes, it is a risk! First you have to learn to 'pass', which some people do better than others. I am lucky; while I was not attractive as a boy, and am not now, I was also very androgynous.
For the first 3-6 months, I got 'sir' more often than 'ma'am'. Every time I went out was a little more humilation on the fire. I came out to my family, risking losing the people who mean the most to me in the world. I started taking hormones, which can have side effects like any other medication.
People looked at me strange. People got hostile. I got called a freak, and worse.
To say that surgery was 'mutilation', or cosmetic, or unneeded, or unwarrented, is stupid. Who would put themselves through the hell you have to go through to /get/ SRS if it wasn't needed?
SRS as a surgery has a very,very high 'success' rate. Almost noone who gets it regrets it. If /any/ other surgery had the results SRS does, it would be called out as a miracle.
Should insurance and/or public health cover it? Yes. Just as it should cover a broken leg, or an STD, or childbirth, or alzeihmers, or cancer.
What, in your mind, /is/ worthy of coverage? You trivialise this with 'subjective feelings of identity'. I call BS on that. You shouldn't get treatment of your broken toe because of your subjective feelings of pain. You shouldn't get treatment with depression or anxiety because of your subjective feelings of sadness. You shouldn't get help with a heart attack, either, since, y'know, it's not the public's fault your heart seized.
Where do you draw the line?
SRS is not trivial. It is not casual. It is not easy. It is not cheap. My transition to date has probably cost about $50k since I started, $30k of that being getting SRS. I'll be in debt for the next half decade at least, and that's not counting the fact that I have 4 outfits that fit (I've lost 60 pounds since transitioning. Amazing what not hating your body can do for you!), or that I still need electrolysis, or that my reaction to hormones as been mild and the only way I will ever have a proportionate figure is probably implants, which again have to come out of my pocket.
Anyway, I've gone on enough about this. I doubt it will make a difference, but I hope this at least shows a couple people that this is /not/ a trivial thing or a bit of vanity or perversion.