False positives aren't inherently a problem. If you have accurate understanding of the sensitivity and specificity of a test, then you can take action to maximise the utility of the expected outcome.
This means, for example, not necessarily performing a biopsy just because something is there (but it's probably benign). The cost of invasive procedures such as biopsies, and risk-increasing tests such as CT scans has to be balanced against their benefits.
Of course, there is no risk with a full-body MRI scan. The machine itself is expensive, but otherwise it nominally only costs electricity, maintenance and operation. If full-body scans were, as you say, routine, the cost of the machine would be amortised and an economy of scale should prevail.
The problem is legal; if a doctor sees something on a scan and doesn't take action, they can be sued. This is the case even if there is negative expected utility from performing the test due to the low probability of an actual malignant condition. Due to this, it is not uncommon to hear doctors say that full-body scans do more harm than good.
Until the legal system starts to account for rational choices based on statistical information, this problem is going to persist.