I'm in Virginia. We don't have to go to court here, just submit paperwork to the County/City Clerk's office where they review, do a background/credit check, and then approve as long as you haven't chosen something offensive. I don't think I even had to write a reason on the court order request.
That is really what a judge is looking at - have you chosen a decent name, is there a possibility you're changing your name to hide your identity, do you understand what all the name change entails, etc. I've never heard of someone in the US being asked for proof, but it wouldn't surprise me if they had disclosed they were transgender. Legally speaking, you don't need a reason to change your name but I could see some judges giving people a hard time.
If you're worried about it, I would put together a transition timeline. Explain what your plan is moving forward, explain why you haven't moved forward yet, etc. For most people, insurance and cost are the factors that limit how quickly they can proceed with transitioning, so I would just say something along the lines of, "My name change would be the first outward step I'm taking in order to transition. Most of the aspects of medically transitioning are prohibitively expensive without insurance, so I'd like to start with the name change so that I might have an easier time finding a job that offers insurance which would help me take the next medical steps."
