Again, don't be impatient with me, because these are honest questions. But aside from certain more technical career fields, what might be an example of a situation where the lack of a firm grasp of something beyond basic math would really leave someone at a loss? What would you need that you couldn't work out with paper, pencil, and an understanding of addition, multiplication, subtraction, division and how decimals & fractions work?

Well, that depends on how much algebra we're talking about here. Basic algebra is nothing more than you describe in your last sentence plus the not-so-confusing leap of substituting one number with a variable. So yes, if you can do those plus a bit of basic logic like realizing that dividing both halves of the equation by anything (except possibly zero and infinity? dunno) will still result in an equivalent equation, you're pretty much set. (And no, you don't have to know the vocabulary either.) A lot of algebra, when applied to everyday life, seems like nothing more than common sense - but I find it only all the more troubling that kids appear to have so much trouble with it in the US.

Teaching a formalized version allows people to see that thourough understanding and logical application of some relatively easy-to-grasp base knowledge will help them immensely when solving some otherwise very hard problems.

I mentioned computer games specifically because nobody needs them and lots of people play them. Maximizing my character's capability in D2 was more fun then actually playing, and now I can work out which perks in Fallout 3/NV or Skyrim I can skip without losing too much oomph. Granted, I could probably figure those out without algebra as such, but it is so much easier when you know and have some practice with it - plus, as I said, for D2 you'd need some crazy math I was not prepared to elarn for a game.

For a maybe more serious example, I started working with Blender, a 3D modelling freeware, and I quickly realized that the best approach to having all the walls and roofs and whatnot of buildings fit together the way I want to, I really need to figure them out beforehand. Lots of it is quadratic equations, which are not really all that intuitive. I could do it all in blender, but then I'd have to readjust dozens or even hundreds of vertices,

My point is mainly that algebra may become useful unexpectedly. Ignorance of algebra is not likely to leave someone at a loss regularly - if they don't have a job that requires it (and they won't think of getting one if they don't know algebra), opportunities to use it will still pop up, but will be ignored in pretty much the same way I ignore the possibility of applying chemistry to cooking. It simply seldom enters my head as an option.

As for teaching it, I have forgotten a lot of math in my life, but whenever I needed or wanted it, I had only to retrace what I once knew, which was much easier than learning it from scratch, pretty much like a basic, confused understanding of 20th century history still gives you a frame of reference.

One point that I think wasn't mentioned explicitly yet is that I find it very important for people in general to know the basic principles of very important things. I advocate media criticism/applied rhetorics and science theory (Oh, and logic. Economy may be a good idea too.) in school for that very reason. So many people have no bloody clue what tricks politicians use on a daily basis, so many people think that scientists are educated guessers. It is important to know where civilizational accomplishments come from in order to appreciate their fragility and make informed decisions. Which, in a democracy, should be self-evidently desirable. Broad education for everyone may be a hard or even (easily) impossible task, but it's well worth it to try.