I’m trans and from Germany. I don’t get what you mean by “when transitioning you have to change your name”. How do you define “transition” then? Who is stopping you from transitioning?
The underlying problem, and you are right there, is that this new ability to self-ID does not touch on any of the laws of anyone’s medical transition. Germany is a country with a more or less working health insurance system that covers most medical expenses. However, although medical transition is definitely necessary for the wellbeing of trans people most health insurances won’t pay for much of your transition. HRT is pretty straightforward to get and as long as you go to enby-friendly endocrinologists you won’t have that much of a problem getting HRT under your old name etc. Speech therapy is even easier to get. But anything like surgery or other medical procedures like laser/electric epilation hair removal are really hard to get covered by a health insurance. In case of gender affirming surgery, you often need to prove the health insurance how binary trans you are and that you really want to transition all the way. With hair removal they usually would pay for it, but only to doctors on a rate that no doctor would agree on.
I know trans people in various stages of their transition and changing their names is not something that anyone had to do. And if you happen to have a gender neutral name, you certainly don’t have to change it.