It's both, really. The degree, and accreditation are fairly paramount when it comes to employment.
Mixed into all of that are also the issues of scope of practice and competency. Much of what the OP described appears to fall within clinical psych, though if he wanted to practice in the school setting, it is much more of a state by state consideration.
As others have stated, going the school psych route would allow practice in schools, but PP would be harder in regard to getting paid for the work. The cash pay market tends to be very competitive and hard to sustain a practice starting out unless you have an in at a private school, where cash pay tends to be more frequent. There can be a lot of “white glove” needs (often non-billable) when dealing with parents in that setting, which can be a hassle; offering non biz hours may also be needed.
There is also the issue of psychoeducational assessment v. neuropsych, as the OP mentioned ADHD and learning disability evals. Once it starts to spill over into congenital conditions, misdiagnosed learning disorders, evaluating for autism, etc....that's the realm of pediatric neuropsychology, which is a looong and very competitive road. I’ve seen school psychologists get pushed into a more neuro role bc of the lack of pediatric neuropsych availability, which can be a tough position to practice.
Competency quickly becomes an issue when clinical psychologists (generalists) or school psychologists start dabbling in neuropsych. My first ~2yr of research training was in peds doing psychoeducational evals in a joint research project with school psychology faculty, and that assessment work was quite different than peds neuro, but it gave me an appreciation of how quickly those two sides can come together bc cases are rarely clean and easily defined.
Lastly, there are a handful of school + clinical (APA-approved) training programs that would get him almost everything he wants, but that road can be tough because there are a limited # of programs that do it, most of them in the northeast.
Attaining an APA-acred internship can be a bit trickier coming from a school psych program, though a combined program could allow an easier path. There are a handful of solid school psych internship training options within APPIC, so the competition can be high for them.
I post all of this for the OP not to dissuade him, but instead to try and provide more of a context for why what he wants isn’t going to be easy and why some of the responses aren’t what he wants to hear.