I've never experienced/used wire-wrapping and I've been into electronics since the 80s as a high schooler. I started with PCBs. Can someone post a video of this in action? Do you need a special board for this? How do you wire-wrap IC chips? And what's the advantage over a breadboard? -- honest questions.
Well, to be honest, wire-wrapping is pretty slow compared to soldering. Even with a motorized wrapper, you have to load the wire into the tip of the tool.* But it's useful for the home lab in certain situations for temporary and/or replaceable connections, attaching temporary headers, making connections between header pins and solder points, or for making lots of connections on a breadboard without growing a huge plate of spaghetti. You would do this by sticking pin headers into the breadboard along the components and then wire wrapping the jumpers. Another place I might use it is for power leads/probes. I have banana plug leads that terminate in header pins. Besides using in a breadboard, the way I typically use them is to attach wire-wrapped sacrificial 30AWG wire to the end for when the most expedient connection to the circuit are small soldered connections. The modified wrap provides strain relief, and when you have broken/cut the wire so many times, you can remove it and wrap a new piece in place.
I always figured the larger gauges were more of a specialty thing, because they have to go over a square pin of the right size/spacing.
*I used to have some old wrap tools that were different, by OK Industries. They were a blue plastic pen with a wire wrap bit on one end and a small spool loaded with wire on the other end. The wire was continuously fed through the back end of the wrap bit, so it didn't have to be loaded for each connection. I got them in an auction, and they didn't come with any instructions, but they came preloaded with wire. I could be totally wrong, but what I gathered is they were intended for use with polyester or vinyl insulated 30AWG wire, like what they came loaded with. There was a little brake switch to hold the wire. You would wrap the insulated wire to a square post, and the pressure of the wrap was high enough to force the conductor to split the insulation to make connection to the post. So the first wrap would be upside down, so to speak. Then on your second wrap, you would engage the brake when done, and your final twist would break the wire. Neat idea, but the wraps took up a lot of post and I didn't find the resulting connections to be 100% trust-worthy without stringent technique. I might still have one, somewhere.