I'm not sure that appealing to chaos theory really makes the answer more acccessible than the not-so-good answers she's trying to surpass. As she says, weather doesn't really work like that; it's just an analogy. But an analogy to an unfamiliar thing isn't really going to help a whole lot of people.

The problem with climate change isn't about radiative forcing causing a large, discrete jump in temperature. It's a fairly classical, non-chaotic effect. Add a little CO2 and get a little temperature increase; add a lot of CO2 and get a large temperature increase -- when averaged over time (a period of several years).

There are plenty of chaotic elements to understand about both weather and climate, but I don't think they'll help most people understand the difference between the two.

A system can have any number of states (variables that store energy). The output of that system is a combination of those states.

Some states can have slow dynamics, while other states have fast dynamics. Is is possible for a system to have chaos present only in the fast dynamics (weather in this case) and non-chaotic slow dynamics (climate).

And that's all there is to it.

It's similar to how sea level tides can be predicted even if there are big waves all the time.