Aren't things already multidimensional?

You seem to be implying that left-right is the natural ideological spectrum. Isn't it more likely that it's just a particular axis that is salient at this time? Is it implausible that another axis will become the center of conflict in the future? What is the left-right axis? Is it (a) the eternal key to all ideology; (b) an axis of conflict that many care about today; (c) the contrast between a pair of alliances that are pretty arbitrary? Is there even a difference between (b) and (c)? Perhaps whether the axis drives the alliance or vice versa?

How can you figure out which beliefs are ideological and which are self-interested?

Finally, I'm not sure if you are making an abstract point, or whether you are talking very specifically about this article and saying that this particular article would be better framed this way.

Let me try that again. You contrast "ideological spectra" with "alliances between interest groups." There seem to be three contrasts. I think it is important to treat them separately. One contrast is one dimension vs many dimensions. A second is the existence of a high dimensional ambient space vs there just being a graph of alliances. The third is the origin of beliefs or policies in ideology vs self-interest. You prefer your description, but if you could only correct one of those contrasts, which would it be?