Mindfulness usually has a object, which will be different depending on the context. In Vipassana meditation, the object of mindfulness is the mind itself. You could say that Vipassana is "to observe the activity of the mind." In that case, it's natural that the mind will wander and your task is to merely observe that activity. In doing so, the mind might return from distraction, but there should not be a forced effort to keep the mind focused. But mindfulness can have different object outside of meditation (mindful eating, mindful listening, etc).

I think there's a few interesting and different ideas coming together here:

1) Training a skill or several skills to increase opportunities for lucidity (awareness is surely one, may be others)
2) Training whichever skill it is that triggers the moment of transition to lucidity (maybe critical reflection, maybe something else)
3) Training whichever skill it is that maintains lucidity and avoids "zoning out" (I'd call this attention, but maybe described differently)

There is some overlap among those, but we don't to muddle them all together. I think Vipassana can improve all of those indirectly. There might be other practices that more narrowly and directly train each one separately.