This is one angle i've come to understand through practice,i'm horrible at explaining things but i'll try my best:

Mindfulness is simply a by-product of Zazen. In the same way feeling good or gaining muscles is a by-product of working out. It is not something you practice. If you have this mindset that you can cultivate the qualities of mindfulness into your practice,that is backwards thinking. All efforts of being mindful,whether its paying attention to this moment, non-judging, or practicing non-striving or choosing to be patient , practicing acceptance, and being mindful of how we treat people etc etc, all these efforts and practices are a means to an end. And that end doesn't sound so bad: we are told to practice these qualities so we could become more Buddha-like or more joyous in life etc.
But what if we are already Buddha?
And i believe many years ago buddha wasn't talking about those qualities as if it's something we should practice like "pay attention while your eating" or " listen without judging", i think he was simply referring to them as the after-effects of sitting under a tree... it's not like he was "practicing being attentive" while eating rice, he just wasn't thinking about the past and future while eating rice because he was too damn excited about the act of eating "Damn...this could very well be the best rice i've ever had!!".
He didn't judge people while listening to them not because he was taking a deep breath and going into "ZEN" mode and taking the higher road, but because he didn't have the time to: he was too damn stoked at the act of being able to listen to another speak.Like as if experiencing it for the first time. He wasn't practicing patience while standing in line to get his bucket full of water either, he didn't need to, he was already just totally comfortable in his skin, and he didn't see himself as a patient person,he wasn't practicing a quality he could later add to his sense of self,there was no room for a sense of self to take the main stage,whatsoever: He was too busy bathing in the act of standing.
There are zen teachers who say that doing zazen brings us close to the state of mind we had as a child, and when you were a child looking at the christmas tree you were overcome with excitement at how cool that tree looked, you weren't practicing mindfulness and you sure as hell didn't give yourself a quality for feeling what you felt. Even if you felt disappointment. You just were.
And Buddha understood this was all because of sitting still.
Find the Gold first and the rest will follow. And when you see it happen, it will strike you like lightning: the miracle of a bad habit momentarily broken without absolutely ANY effort on your part.
The truth is,in this moment, once seen, there is no room to "practice mindfulness', not even close, you'll be too dumbstruck by reality to practice anything.

Thats my 0.02 cents.