illustration (attribution, if any possible, is at the end of the article)
More often than not, and sometimes due to an utilitarian perspective on meditation (done in order to get a benefit), what most people are aware of while meditating is what's 'wrong': what disturbs their meditation, what disturbs their peace, what tendency shows itself as obsessive and constantly calling for attention thus impeding their freedom, etc.
Cultivating of awareness definitely has something to do with 'accepting' these observations, not fighting them off, just letting them arise, endure and cease…
But cultivation of awareness also includes noticing what's 'good': the fact that we did catch ourselves drifting (even if it was after a long while: it's much much better than 'never' catching ourselves!), the fact that some peace is present, or the fact that obsessing about something for 20 or 30 minutes has actually provided an insight about something to try and do (i.e. a 'solution' might be in sight now, even if peace is not perceptible), or even the fact that we simply did meditate (which might in itself be a first victory into disciplining ourselves into doing what we choose to do, rather than buying into excuses not to do it), etc.
In short, 'awareness' can very much be 'rejoicing', 'appreciating'… Sometimes, appreciation can simply be about being alive and having an opportunity to at least try and cease dukkha: not everybody is this lucky!
It is often said that awareness should be non-judgemental, but our tendencies to judge and discriminate are quite deeply ingrained. So people may struggle to immediately meditate without judgement.
To cultivate awareness of "good and bad" (rather than just the critical thoughts about what's bad) is a gate into a more equanimous meditative state.
If your brain is good at discriminating but is unbalanced in what it discriminates, may I suggest you meditate not by 'repressing' discriminations but by 're-orienting' the tendency to discriminate: work on discriminating everything, not letting biases perturb your ability to see.
Be present to the 'whole' experience, not just one side of it.
If you catch yourself drifting one side or the other, don't repress the drift by 'forcibly' aiming for a Middle…
You might find yourself constantly coming back to the Middle but also constantly drifting again to the same 'side': that's a harmful tendency (typical of e.g. depression)! Instead, cultivate seeing the other side (and this may mean enquiring into what's 'bad' while you experience bliss, i.e. not buying into peaceful apathy)!
When you catch yourself, don't aim for the Middle: voluntarily look at the other side, and only later come back to the equanimous Middle (anchored in your focus of meditation, be it the breath or anything else)!
Cultivate a 'wider' perspective, rather than a 'narrow' Middle…
When the drift from the Middle randomly is to one side or the other, instead of systematically being to the same side, you've progressed a lot and conquered a harmful bias, even if you still drift often ;-)