Can't comment on the FAA rules but in the UK there was always a requirement for an IN FLIGHT VISIBILITY for circling AND also a minimum reported vis/RVR for the runway on which one intended to land.
For Cat C the IFV requirment was 3700 metres and (typically) 600 metres RVR (or greater if no approach lights etc) minimum rvr for the landing runway.
So for the FAA minimum is the 2 statute miles the minimum IFV for the circle but the 2.3 miles the protected area for circling?
In the company I fly for we circle on a regular basis but have structured procedures for doing so and are well trained for all the options and how to carry out a missed approach if visual reference is lost. Circling is perfectly safe so long as you obey the rules.