First off, the fact of the matter is that the technique described on the website was described badly--it isn't a good way to teach it, it's more than a little self-contradictiory, and there's too much theory that isn't very good theory. In other words, the Delayed Sword described is inaccurate, a teaching problem, and bad ideas.
Second off, the fantasy of being perfectly, zennily aware at all times. Does this include, say, in the toilet? C'mahn.
Third, yah sure, increased awareness is good. But this, "combat awareness--" well, I've been reading Frantzis' stuff recently, and he says something I've suspected for a while: such constant alertness is actively dangerous. It's bad for you to be on guard all the time; beyond the philosophical issues, being on guard all the time means a constant physiological response to trouble that isn't even there. I agree with the last several posters: awake yes; alert---nah. We're not in a wartime jungle.