It seems like you’re promoting a self fulfilling prophecy. Doctors aren’t good at reading scans, so scans will cause false positives.
But doctors are capable of learning and improving their skills. They will surely rapidly learn to screen a lot of false positives, thereby saving lives.
"More frequent scans per patient" != "More scans seen per clinician". You would just end up having to hire more clinicians.
(Yes, your family doctor might end up seeing more scans in this scheme, but it is implausible that they would be better at evaluating them than a trained radiologist, which is who this would almost surely be deferred to anyway.)
Relatedly, for many types of brain tumors, you simply don't know what exact kind of tumor it is until you actually do a biopsy (= you undergo surgery). This can be important information ("how aggressive is it?"), so in some hospitals it is common practice to send a tissue sample for identification and get back the result while the surgery is still going on.
Mammography is a really good candidate for random screening and it's still debated because of false positives and the impacts of misdiagnosis.