How important is the dimensional accuracy of filament relative to the detail of a print?
Dimensional accuracy is not as important as dimensional uniformity. I can print with undersized (or oversized) filament, adjusting the flow appropriately, provided the filament has a consistent diameter. When creating filament in-house, without expensive equipment, it is difficult to maintain the same diameter throughout the entire extrusion. It is likely this extrusion diameter (when creating filament, rather than the output of the actual print head) to which Barafu is referring when he mentions his tolerances: +/- 0.05&nbsp;mm in diameter. Which is reasonable.

The "miniature printing" comment likely refers to printing miniature models for tabletop gaming.

If the source filament becomes wider than expected, the output will have overflow, or more material than desired will be deposited, and this will certainly affect the quality of the piece.