With regard to psychologists, I believe the education departments in each state do have psychologists, maybe. They might even do individual educational assessments on children sometimes but I can't think of ANYONE I know who has a child that has been assessed by an education department psychologist. I think their role is primarily to advise schools from afar (ie phone and email) about reports already done, or managing behavioural issues and decision making. I am not actually sure what they do, but I believe they exist...

Private schools routinely have school psychologists, I don't believe I have ever heard of one doing an IQ assessment, they seem focused on counseling (ie social, emotional, behavioral and usually short term)...

Educational assessments are recommended by schools, public and private, and then you are expected to organise them... Wealthier families will just pay (possibly after a 6-9m wait). Other families will go onto a very long waiting list for a public hospital based assessment (more likely 2yrs), which I think are probably also only available if there is also a developmental or behavioral issue being addressed (ie not an option for a supected gifted child with suspected dyslexia). In some scenarios it might be possible get medicare to help pay for a private psychologist IQ assessment if a pediatrician has requested a psychologist assessment for a developmental issue and the psychologist uses and IQ test as part of their assessment... We did this once, but I am no sure if you can still do it, even so there is still a decent out of pocket.

Given that any time you change schools, or states, or maybe it's just been 2yrs, or now you want yr12 exam provisions... You will be asked to do this again and again. I have at this point paid for at least eleven full ed psych evaluations. And I am not sure how many OT evaluations. What is really fun is that it is very common that psychologists who like to do assessments will not want to engage with school advocacy at all, beyond some recommendations in the report. Or they will charge by the hour if willing (I have never succeeded at actually having a psychologist who had done a test intervene with a school directly).