I wanted to weight in, as I see some incomplete or incorrect information that I have seen on these boards before. I come from a School Psych PhD program, and am finishing my internship this summer.
I frequently see confusion about what school psychologists do at different levels of training. The Educational Specialist level (EdS, though in some states it is an MA or M.ED) is the traditional training for school psychology. These are the folks who typically work exclusively in schools doing special education evaluations and IEP stuff. In the last few decades, the role of this position has expanded a bit to include systems level support (RtI, PBIS), as well as an increasing focus on behavioral consultation and academic interventions. At the EdS level, school psychologists are trained primarily in assessment and consultation, with a sprinkle of treatment and counseling (of course, different programs have different emphases).
A the doctoral level, school psychology becomes more similar to clinical psychology. In addition to the above, school psychologists receive extensive training in providing evidence-based intervention (CBT, ABA, etc.) and much more in depth training in clinical assessment. And of course in conducting research. Doctoral level school psychologists do often work in schools, but are more commonly seen in private practice or hospital settings doing assessment and treatment of youth with mental health needs.
So, school psychology is rather distinct from clinical or counseling psychology. There is much more emphasis on systems level consultation and academics, and less emphasis on providing therapy. It would be inappropriate to have school psychologists train as clinical psychologists and then specialize on internship; the coursework and training throughout the program is different.
I understand the frustration regarding EdS level school psychologists having the unique ability to have psychologist in their title. Personally, I think a different term would be more appropriate (Texas calls them Educational Diagnosticians, for example). However, it remains critical that the EdS level remain as the entry level for school psychology. There are already positions going unfilled across the country, and requiring a doctorate would further decrease the availability of services to kids.
To the OP: if you are interested in specializing in school-based services or assessment, a PhD in school psychology sounds like a good fit. You can still provide therapy with this degree, but your program will likely have less emphasis on this than a true clinical program. If you want to specialize in therapy, but still have the chance to work in schools, then the clinical route may be better. A PsyD in school psychology is an option too, but be somewhat cautious. While there are strong PsyD programs, there are very weak ones too. The match stats for unaccredited PsyD school psychology programs are among the worst in the business.