I do not take wu wei to be apathy. What you describe as "giving up purpose" I would describe as "giving up ego." That doesn't mean there is no purpose. The purpose is contentment. harmony.
Hmm. Perhaps giving up purpose is equivalent to giving up ego but they're different flavours of the same detachment ice-cream. Uh, my metaphors are awful.
But anyway, the thing is I really feel it makes no sense to ascribe purpose to Tao, including contentment and harmony. It just so happens that giving up having the purpose of contentment and harmony tends to make those things more likely. So you see there is a
massive difference between our positions.
More verbiage to explain my thinking: Giving something up to gain it, or something thereabouts is a classic Taoist principle and bares some relation to the "invest in loss" technique of Tai Chi Chu'an. Several people I've known who are fascinated by Taoism have either talked about explicitly or employed a kind of technique that strikes a chord with this. They either fell into it by accident or cultivated it over time and namely it involves dissolving purpose, meaning, value, god(s), decision making, etc. in such a way that they sink into the subconscious where they slowly coagulate together in ways that aren't possible by holding them up in the conscious mind.
Hidden from view (obviously this is hypothetical) but nevertheless potentially immanent as intuitive impulses and impressions the Taoist cultivating this way learns to follow (or rather, fall into) the 'mystery' and, with a diminishing sense of control & focus, often finds that their life becomes more and more integrated with deep a sense of meaning and purpose emerging out of that regardless!
A classic outward symptom of this approach is to find that all one's plans, efforts and thoughts go absolutely nowhere (and quickly!) such that it seems like there is nothing but failure (even if one has already chosen to 'drop out'). In fact it can make someone feel very apathetic to begin with! Why bother, hmm? But when one looks back at what has actually taken place in its possible to see that great opportunities arose, skills blossomed and things changed for the better where, when and with an ease that one least expected them to. This is Te: mysterious and dynamic virtue/power of the Tao.
It might sound a bit crazy but I swear I've seen many people going this route and also had enough experience of it myself. Its a real path and is similar in some ways to Buddhist approaches but has a uniqueness to its philosophy that is very much Taoist.
One short snap of what its like is when we feel that nothing we can do, or can't do, will make any change to our sense of worth - even that everything we do or say is pretentious somehow. Following from this joy and goodness come welling up for no reason at all.
No reason, no purpose, no meaning and yet having the experience of all three of these simultaneously without trace. That's what its like. To put it another way its the experience is of finding oneself to be a better moral being after letting go of any notion that one should be. Its a gamble - that not trying to be good will actually turn out to be good. To me its about here that the distinction between apathy and apophatic mysticism can be found: apathy doesn't usually require enormous commitment, concentration and courage in dropping social expectations and personal desires.