Your approach is as abstract as the excerpts of Thurston's remarks that you quoted in the other thread. This is ironic because the theme is to exhort mathematics instructors to pay more attention to intuitive and applied methods. However, I can't picture what audience or situation these articles address. Is it secondary school teachers? High School teachers? Undergraduate instructors? Something you do at your school?
The list of applications of group theory is impressive. I think college instructors are already aware of these facts. As to what high school teachers could do with that material, I suppose they could make those verbal statements. I don't think either of these groups of teachers would have time to go into the details of those topics. That is the one of the problems in finding a way to motivate group theory. The applications (that I know about) have great complications besides the theory groups. Trying to study the applications is too exhausting a digression.
The best way for me to imagine the audience for the paper is to pretend that it speaks to the authors of mathematics books. I grant that this is an important audience.
I have a strong reaction to the idea of introducing groups as automorphisms of polygons. I can speak as an expert on this topic. When I was a teenager, I began reading a book called "Groups And Their Graphs" by Wilhelm Magnus and Israel Grossman. (These was in the years just after Sputnik and all sorts of mathematical literature oriented toward kids was being written so the US would not fall behind the Soviets in technology.) The first part of the book upset me. The mapping of polygons upon themselves was presented with vivid language like "rotate 90 degrees clockwise". But what went on the book bore no relation to such a dynamic process. There wasn't any motion or time involved. Apparently you supposed to pretend the rotation happened instantaneously - Zip! and it's done. I got used to that. But then there was the problem of why the book never talked about the edges of the polygon. It only talked about which vertices went to which vertices and never about what happened to the points on the edges. I finally decided that there must be some unstated convention about the edges. Maybe you had to assume the line segment between the two vertices went along with the vertices. So I developed a slight resentment against this method of talking about groups. I felt the authors had it in their mind to talk about mapping discrete sets of vertices to themselves and they weren't really interested in the way that figures are really moved around. They were also pulling the typical algebra book stunt of expecting the reader to adopt certain tacit assumptions about the situation without plainly stating them.
This book emphasized the "generators and relations" approach to groups and the later part of the book had more interesting graphs where the vertices were elements of the group. I didn't have any problem with that kind of abstraction. The picture of how that kind of graph relates to the material on mapping polygons to themselves was not clear in my mind - and still isn't.
So, with respect to using the automorphisms of polygons, I think the presentation must explain why the discussion focuses on the vertices. It also must explain why the student need not worry about the edges. But we must worry about the edges, otherwise why not map the set of vertices onto itself in some crazy manner and call that mapping a symmetry? What is really going on with those discussions of polygons anyway? It must be some sort of quotient group of a Lie group.
For me, accepting the definition of a group never seemed like a big problem. Some other book made it clear to me that a group is a nice set of functions that is closed under the operation of composition and taking inverses. (It surprised me by giving an example where the group elements were simple algebraic expressions.) This makes non-commutative behavior clear. The fact that the functions are automorphisms seems to be a simple consequence of the desire to compose them. You have to have them operating on the same set in order to do that.
I think the problem of how to present an intuitive foundation for elementary group theory is unsolved. The scope of the problem is more than just motivating the axioms. It's also explaining why a person should care about things like normal subgroups, Lagrange's theorem, Sylow subgroups etc. (I, myself, still don't know why I should care about Sylow subgroups etc. )
I'd like to hear people's ideas about this. The most important problem is how to approach normal subgroups.
The paper's audience is unclear. Something between curious undergrad, professors and book writers. Maybe I was writing it to myself so that I could put my ideas in writing.
I totally agree with many of your claims - Why vertices and not edges, how do we continue to motivate normal subgroups, etc. I do plan on writing a complete book or something of the sort developing a large chunk of elementary group theory by this paradigm. It should end up much more tangible than the paper linked here (which I agree is written abstractly). Maybe we can collaborate on this? I'll be quite busy in the upcoming two months or so but after that I would like to start some sort of new form of collaborative web system. This system should allow for whoever wants to cooperate on this by asking questions and/or suggesting answers to how certain ideas might be approached.
I'm interested in such a project. In two months I might even have fast internet service. (Today's local paper, the Sun News, says that Qwest is feuding with my internet company and threatens to cut off all its subscribers.)
You may be interested in my DC Proof software (free download at www.dcproof.com). You can bring up the axioms of group theory (or ring theory or Peano's axioms) onto the main screen and "play" with them using the rules logic and set theory, which are built into the program. It is a purely algebraic approach. I include a proof of the uniqueness of the identity elements in the Samples directory.