Why is the initial first-line response to a psychological problem not a brain scan? I get the impression most people having a cognitive problem are encouraged to see either a psychologists or psychiatrists (the actual medical doctors) to
engage in verbal or pharmaceutical therapy. I actually knew someone who, in the '80s, was having visual hallucinations for months and seeing a therapist before she actually got a brain scan and discovered it was a benign tumor.

While I can understand that scans cost money, since repeated therapy sessions also cost money, I don't really understand why the prevailing advice isn't "scan the person's brain first, and check their blood, then try chatting with them if it isn't obvious after that". Wouldn't you want to go with the obvious, empirical objective measures first, and switch to heuristics after that?

Which isn't to say that it isn't important to have an initial conversation with the person- for the same reason a person should have a conversation with their GP- but it seems like that only makes sense as a precursor or supplement to the physical testing that also needs to happen.
The first stop when a non-emergency medical issue evolves is, as you say, a visit to the GP. This is not a supplement to diagnosis. Instead, a visit to the GP can be a critical step in the treatment process and a GP will assess what the most probable underlying cause is during a consultation, based on a physical examination and verbal communications. Then the GP can refer the patient to the proper next medical practitioner for specialized help. Note that a diagnosis involves mainly excluding other causes and narrowing down the list of options.

Your case report lacks some critical information (e.g., age at diagnosis etc.), but the statistics behind the diagnosis can still tell a lot about the GPs decision: 


The incidence of brain tumors is about 22 cases per 100,000 individuals (0.02%). Brain tumors are most likely to arise in kids or late teens. Dependent on the age of your friend at the time of diagnosis, this may have placed them out of that age window.
The incidence of visual hallucinations is about 17 per 1000 (1.7%) (Tien, 1991), i.e., orders of magnitude higher than brain tumors. In other words, just a minor fraction of visual hallucinations are caused by brain tumors. A more likely cause for hallucinations are positive psychotic symptoms, the realm of psychotherapists and psychiatrists. Most visual hallucinations are caused by psychosis (e.g., schizophrenia), Parkinson's disease and Alzheimer's disease (source: WebMD). The latter two are typical cheriatric diseases (diseases linked to old age). If your friend was not considered an elderly at the time of diagnosis, it would make psychosis the most likely candidate cause of the symptoms. 


Brain scans do not diagnose Alzheimer's or Parkinson's, nor psychosis. A brain scan would only, perhaps, find a tumor that has a very small chance of occurring. Moreover, a single scan may not be enough. The tumor may be small and clinical tests, fMRI, PET, or bone scans may be necessary (source: JHMI), which can result in an extensive list of scans, many hospital visits, and associated financial burden. The GP chose the best hit, and that's a psychiatric issue.  

Reference
- Tien, Soc Psychiatry Psychiatr Epidemiol (1991); 26(6):287-92