Firstly wire-wrap is horrible to do, plays havoc with the eyes staring at 100's of sharp points all the time.
The pins must be hard-square profile with pointed tips and gold plated (to ensure a gas-tight weld), the correct wire and tool must be used (wire wrap tools are high precision and very expensive). Wirewrap wire is a specific thickness, coating and insulation material, all the mechanical properties matter for reliable creation of joints. Long pins are standard to allow upto 3 wires to any pin, which is pretty much a requirement for many circuits.
Wire-wrapping typically adds a lot of stray capacitance to circuits, unsuitable for modern high-speed logic for instance. Its hard to make a ground-plane with wire-wrap!
PCBs, breakout PCBs and breadboards, IDC connectors and ribbon cable, etc are much to be prefered - these days that's the easy and cheap option - that's why wire-wrap died out I think.
However with the correct equipment a wire-wrapping robot is probably still the most reliable way to make circuits, well, low speed circuits, nothing beats a gas-tight weld, and tin-whiskers unlikely to be an issue with everything on 0.1" pitch.