I suspect it is not actually stiffness you want so much as a shit-ton of drag !
Here is a rope made from a helix with just 8 points (Uniform, 8 interpolation) falling from horizontal to vertical very slowly and without stretching.
Rope Unfurl CBR 01.c4d
The key settings here are:
Very few points in the source spline gives ropes that don't stretch so much. Scene / simulation / Substeps also affects this, so mine is bumped up to 40 or so. Other than that it's mainly high drag value in the rope tag itself that chills this out enough to make it fall slowly and smoothly with that unbending motion you mentioned. If you want it even slower, that can go up more, as can the global drag in Scene / Simulation settings, though note these settings are cumulative so it is easy to overdo it. Thereafter you can get some wind or turbulence forces in there to make it swing once it is hanging vertically.
CBR