Wirewrapping is quick and cheap.
You can get lots of connections into a very small space.
It withstands vibration well.
Components are re-usable.
However, it is difficult to make it look neat.

Fig. 2 shows a regular wrap.

Fig. 3 shows a modified wrap, where some of the insulation is wrapped around the pin.
Modified wraps are more vibration proof.

There are many types of tool available, from a few pounds, up to several hundreds.
All usually have some means of stripping the correct length of insulation.
Separate special wire stripping tools are available.

When wrapping, push the wire all the way into the tool.
Allow the tool to rise up the pin under its own steam.Don't pull or push it.
Keep the tool on the pin until the wrap is complete.

Avoid the following:
Turns wrapped on top of other turns.
Spaced out turns.
Pigtails, where the end of the wire is not wrapped but sticks out.
Loose wraps i.e not tight around the pin.
Insufficient turns.

Looking at Fig.4; if it is necessary to remove a bottom link then two upper links must be removed.
If you have made wraps as in Fig.5 then you may have to remove ALL of your wrapping. PLAN AHEAD. Don't change levels.