I'm not sure if I 100% get where you were going with that, but I can tell you that Te is a way of life for Te-users; you can't put your Te on before you leave for work, and take it off to hang up on your coat rack when you get home. (I find that that sort of function attitude makes more sense with inferior functions, e.g. my INTP dad definitely puts on and take off his Fe at will.) The way people describe Te does make it seem more business-related, you're right, but that's just because business language has irreparably seeped into MBTI descriptions (probably because of how frequently the MBTI is used in the workplace, vs. in other environments).
Take my experience on this thread, as an example. I entered it because I was essentially given a task by Rex and SD: Use your expertise/experience and help the other TJs make Te clear for the Te-uninitiated. So I came in, methodically responded to just about everyone in the thread in chronological order (to complete the task as thoroughly and excellently as I can), and my on-topic responses have all been formatted thus: Thesis statement. Bullet points: 1) 2) 3). The End. No frills. Like others have been saying: Get the task done as efficiently as possible.
And the thing is: this is all pretty much stream-of-consciousness. I don't have to work hard at all to operate like this. Meaning, the way I think really is
that direct, a lot of the time. Yes, I go on Ne tangents sometimes, and yes, I do tend to feel like my mind is much more chaotic than my interaction style would lead you to believe. But the fact that I'm entering this thread like "Here is my input. Here is why it makes sense. Boom, boom, boom, point A, point B, point C. Done." is a pretty excellent example of "mundane" Te, i.e. Te in everyday life. And you can likely see, from how I phrased it, that it is VERY similar to workplace Te, just funneled into a different environment, with different goals laid out for it to accomplish.