When is the AF-assist light actually necessary?

AF assist light does not affect picture qualiyy per se.
What it does is assist the camera to focus by increasing light level.
Once focus lock has been achieved the assist light has done its job - its light does NOT appear in the image.

Experience will tell you when the light will improve focusing.
Often the camera may attempt to use the light when you can focus adequately without it by using a little extra effort. As light levels drop contrast levels fall and focusing becomes more difficult. By focusing on an area of visually distinctive contrast change you can focus in such situations and then recompose. I almost always waste all the nice focusing sensors that modern SLRs provide and use centre 'spot" focus plus recomposition.

A point will be reached as light falls where focusing slows dramatically or becomes very unreliable. At that stage you are well past where the AF assist light would have been of some use.

When shooting in good light levels a camera may 'decide' to use AF assist - often due to a darket area somewhere in the scene. So, even in the brightest day you can not be certain that the AF assiste will not be auto-triggered. In "street photography' or when photographing crowds, fairs, shows etc it is often not useful for people to be disturbed and alerted by the visually bright AF light. I've spent quite a lot of time wandering around china with a camera in recent years. I find that eg Chinese policemen are not keen to be photographed at the best of times. Alerting them to your artistic attentions with a bright red lights seems unwise [tm].

I do my best to try to remember to turn AF assist off when I am in walk around mode :-).