I agree with most of the comments. Potential for severe crowding, potentially cold (wear long underwear!), potentially very windy, potential for rappel epics, long enough to end up benighted. Overbolted at this point although not originally (I've done it twice about fifteen years apart), way easier than comparably-graded Gunks climbs, but of course much longer, so multipitch efficiency (not exactly the hallmark of many Gunks climbers) matters.
I don't think the actual climbing is memorable, but the formation itself and the positions are spectacular, and there aren't many routes in this country of that length and steepness with climbing so basically moderate. So overall I think it deserves its classic status, in the same way perhaps that High Exposure, with its rather uninteresting climbing, is nonetheless a classic. At nine pitches, there is no comparing CC to Dark Shadows (unless you do Dark Shadow to the top of Mescalito, which only a minority of the ascenders do). I agree that the climbing on Frogland is more interesting (and a bit harder), but the ambiance on CC is superior and it is longer. Jubilant song is a very good route, but has two pitches that are harder than anything on CC. One of those pitches is 5.9 in my opinion, with the pro beneath your feet at the crux.
The second time I did CC it was in ominous weather conditions in March, and we had the route entirely to ourselves. It was dark and chilly, and the wind gusts were strong enough to nearly blow us off our stances on occasion, but the compensation of being alone on the route made up for the extra difficulties. (I should add emphatically, however, that one should get off and stay off the rock---for as much as a few days---if it is soaked by rain. In our case, precipitation never happened.)
I think the somewhat dicey rappelling situation is best handled by climbing with half ropes. There is a real possibility of hanging ropes on most RR climbs, and if you aren't there on one of those crowded days when there are others above you who can help, you can be in for a serious epic.
This is especially true if you use a tag line and set up your rappel in the usual way, which involves pulling the tag line. If the main rope hangs on the way down, you are left with nothing but a tag line, at which point you are just a dumb tourist stranded on a cliff. If you insist on using a tag line, then learn how to set up your rappels so that you pull the main rope. Of course, this means you have a very thin tag line falling down the rock, and it is probably more likely to jam than a normal-sized climbing rope.
If you are rappelling in high-wind conditions, you need to have your techniques for that dialed---do not throw the ropes! I think the best method is to lower the first person down, but you can also have the first person carry flaked coils with them. If, like me, you use a rappel device that tends to twist the ropes a bit, then carrying flaked coils, in addition to the tangles that sometimes happen with the system in any case, can be pretty slow and clumsy, because there is no place for the twists to go to unwind.