It seems clear to me that that which is aware of sadness, anger, or anxiety is not sad, angry, or anxious. But shouldn't this work the other way as well? Wouldn't this mean that that which is aware of happiness or joy is not happy or joyful? It feels like by this logic that mindfulness, when applied to positive states, nullifies them or diminishes them and that one wouldn't want to do this. If mindfulness diminishes the power of negative emotions wouldn't it diminish the power of positive ones as well? I'm not sure how this is desirable. What exactly am I missing?
You are correct in one part and making a faulty assumption in another.
You are correct that observing mental/emotional states with a kind of scientific detachment puts the subjective point of view in a position where pain gets less painful and exaltation gets less exalted. Buddha mentions this method in the suttas.
At the same time you make a faulty assumption that your idea of happiness is an obviously desirable state. This is perfectly expected as most people have never really thought about and don't have a clear definition of what it means to be happy.
Buddhism differentiates between a worldly kind of happy states - joy, comfort, pride etc. coming from achieving success, wealth, fame, etc. on one hand - and "spiritual" happy states (jhanas and their more casual precursors) on the other.
Now, what are these "spiritual" happy states? In Buddhist definition they are all approximations of Nirvana. "Approximations" in the sense that they are of the same essence as Nirvana, just not as complete and refined.
I don't want to get too philosophical with the subject of Nirvana here. Let's just say for the purposes of this discussion Nirvana is like an infinitely refined and perfect Peace. The technical term for this groundless boundless super-peace is "suchness" (tathata). Which clarifies the "spiritual" happy states as various approximations of the super-peace.
Now we have all the concepts we need to answer your question. If you contemplate your mental states with detachment, resulting in the worldly states fading out and ceasing to be, whether they are unhappy states or happy states, the remaining residual will be some kind of partially-peaceful state gradually approaching the super-peace of Nirvana.
To a worldly-oriented person this probably sounds scary or sad: "oh my god, I'm giving up happy states". But for a practitioner of dharma this is perfectly rational and logical: by definition all states except Nirvana have an element of "this thing over here is not perfect, i wish it were different" in them, and certainly worldly happy states are full of such bitter elements, especially after the first wave of exaltation is over. So why wouldn't I want to contemplate my worldly happiness (and even an intermediate "spiritual" happy state!) with a detached mind, knowing that the resulting state will be more satisfactory due to having fewer inner contradictions in it!
To put it simply, why would I cherish a turbulent, fleeting, partially imperfect instance of worldly happiness that won't last anyway -- if, exactly by giving that up, I can open myself to the perfect non-fleeting super-peace.
To answer your heading question: "that which is aware ..." Here, (1) 'that' which is aware is awareness, awareness of mental states. (2) 'What' it is aware of are mental states (for example, fear). Awareness of fear is neither the same as, awareness in general, nor the mental state of fear.
Awareness is not what negates or nullifies the anger or anxiety or any other negative mental state. It is required of course. But some thing other than mere awareness is required to nullify a negativity- That is the analytic side. That is only possible when awareness is established. (In the analytical reaosning, seeing phenomenon as empty of self does this diminishing and in the analytical meditation (vipassana) the establishing of equanimity over vedana does this).
When negative mental states are thus analysed in meditation, then by the very nature of them they get nullified. But the nature of positive mental states isnt the same. (Hence also why they are categorised as positive or negative). The removal of negative defilements also increases the positive states.
Being aware of this does not diminish happiness or joy because one, awareness does not diminish them, and two, it is not in the nature of happiness or joy to be either diminished by the mere awareness. Then the analytic meditation also does not diminish them. It in fact enhances them even further.