Its the need/want to go outside the protected area for the various cats.
CAT C is radius 2.7
CAT D is 3.6
CAT E is 4.5
With a MDA below 1000ft.
I have been challenged about staying in the 2.7 which makes getting the stabilisation gate shall we say interesting. Some SOP's have a partial stabilisation gate at 1000 for circling and 500ft for complete. Others they don't even mention it.
in theory the vis minima is 2NM for CAT C and 2.5 NM for cat D I have never tried a sim approach in viz that low but I really can't see me being able to keep the runway in sight and be stable. And 500ft would be challenging wings level etc etc. And that's with no specials
Flap 0 and Vref of 170knts and bank of 30 degrees your not going to make the turn to finals with 3.7km viz and field in sight. 6.5KM viz of cat E you might have a chance. Stabilised borderline. Terps cat E is 3.2 km viz.
Most CAT C circling approaches I have seen and been tested on never seem to stay inside the protected area if you do them the way the TRE wants them, in fact most are bordering outside the CAT E protected area.
These are just thoughts on what I have seen on circling approaches over the years. I could get the plane in but it would be using "cowboy" skills which thankfully are extremely not current these days, and the safety officer would quite rightly be sending me to the CP for a right roasting. Those cowboy skills are not trained for in modern CAT operations it would be a 10 deg AoB constant aspect, roll the wings level at 200ft and land.
A circling approach at min viz and minimas for me is a last ditch get it on the ground in real life, no more escape options left. In the sim its good for training and making you think in contrived situations.
Real life I have had to operate into single ended IFR approach runways and done Circling approaches for real in a CAT B machine but without modern stabilised approach SOP's I would say all of them would have been categorised as unstable, runway centre line and bank angle over 5 deg not obtained by 500ft. . My current C machine I wouldn't even attempt it unless we were going to crash otherwise.