I wonder if we had constant moment to moment awareness if we could even function as we do now. I think there would be sensory overload or a white noise masking effect, and although the experience could be more enjoyable to most initially I think it would be subdued over time. Stimulation/appreciation/awareness is best served as a periodic sample or a contradiction of the norm. I don't just mean in a "familiarity breeds contempt" sort of way either, even though most do realize the treat a few times a year tastes better than the treat twice a day. I know everyone seems to think a bit different but that is sort of how I feel it would work with me.
Although I tend to agree with the sensory overload possibility mentioned, there may have been somewhat of a misnomer in the caveat of "conscious moment-to-moment awaress". That is, such moments could be either discreet and connected, (as close to continuously aware as possible), discreet and unconnected, (being consciously aware of separate moments which aren't directly connected), or some sort of mixture of the two. The first probably would lead to sensory overload, (or at least, insomnia), although that would depend upon how much input overloads an individual consciousness. The second possibility could be covered by shifting some awareness to the background, as mentioned below. This would include both semi-conscious awareness and subconcious awareness, (but not unawareness, by definition). The third possibility is likely the one most people use on a daily basis; a mixture of connected and disconnected moments of awareness, (with aspects of either manifesting to varying degrees).
I have developed a habit where I can work on problems in the background of my mind where I can be always aware of them and apply a bit of conscious thought to solving them but basically I keep them in a 'foggy' state and revisit with deliberate dedication periodically. If I focus on them too closely I lose the background solving automation and have to do it all deliberately. If I do it just right though the answer is solved for me and I can realize it almost as if it were given to me by another but since it is from my thoughts it isn't foreign and the solution (whether correct or incorrect) is completely natural to my understanding and thinking and required as much active thought as typing would to an experienced typist. I realize this isn't a direct contradiction to moment to moment but it is given as an example of something I couldn't do moment to moment and still manage other serious thoughts moment to moment.
Most likely, few, (if anyone), maintains a constant and continuous moment-to-moment conscious awareness. It is possible that this could be managed to an extent from time to time, (discontinuously), however. For instance, we possess certain autonomic functions, (heartbeat, breathing and other internal biological functions), of which we normally aren't aware of on a conscious and moment-to-moment basis. That's probably a good thing, otherwise we'd likely to overwhelmed by purely 'system maintenence' functions. Being consciously aware of the _process_ of internal thought is another matter. This differs from being aware that one has thoughts and may be more than a recursive awareness of physical synoptic functions. Moreover, conscious awareness can, (and does), extend beyond an awareness of 'self' to at least a partial awareness of that which is not of the 'self', (external environment, other people and other concepts for instance). Such externally-directed awareness varies in degree and one could posit the aforementioned sensory overload in this regard as well.