I agree that, in an ideal future utopia, the removal of gender stereotypes would solve a large proportion of the problems. I don't think we should remove the concept of gender altogether though as, societally driven or not, most people do derive some of their sense of identity through it - I think? Or maybe we don't, I don't know. I just know that I could never question my femaleness - but maybe that's just a sex identifier, not a gender one.
For now, in the current world, I think the current system you describe, OP, is quite a good one. It allows people with genuine sex dysphoria to find a way to live their lives but is lengthy and intrusive enough to mean that only those who are serious about it will go through with it.
To be honest though, I am not convinced helping people to go as far as they can towards a biological impossibility is a good use of funding. I know it's controversial to draw parallels with other MH issues and maybe I am transphobic, I don't know. But I do see gender and sex dysphoria as mental health conditions.
To draw an analogy which works in my head (but maybe not in practice) - I have anorexia. Although expensive, I think it's right that I should be able to access NHS treatment to help me address body image, body dysphoria, gain weight, anxiety, find coping mechanisms etc. I do not think I should be able to access NHS treatment to weight loss aids, weight loss diet advice, liposuction, plastic surgery etc.
In the same way I think someone with gendery or sex dysphoria should be able to access NHS treatment to help them address self esteem, bosy image, identity, dysphoria, depression, anxiety etc. I am not convinced they should be able to access NHS treatment to physically or chemically alter their bodies. II'm not sure it's the best way forward to treat it in such a different way to how other mental health conditions would be treated (ie give what the patient feels will fix them rather than treat the issues behind the problem)