"The major open question regarding non-Newtonian fluids is very simple: what equations should I use to model this fluid?"That pretty much sums it up nicely, doesn't it? Note the first word: "The". Not "A major open question...", but "The major open question..."
The answer of course would be the simplest model that captures the essential features that you are interested. Sometimes an empirical power law will do: η = m γn-1 (that γ should have a dot over it, but that is beyond my HTML skills) and sometimes you pull out the 8-Constant Oldroyd model (which is far beyond my HTML skills, so you can Google it if you are interested). Sometimes a non-objective equation  will be just fine, sometimes you don't care about normal stresses, and sometimes you have to bite the bullet and go all in.
This still doesn't answer the question of how do you know which model to use. I admit that I am dancing around the question as I don't have a good answer. There is no cheat-sheet that you can refer to. I've not done much modeling over the years, but the advice that I would give would be to look at what others have done with similar type problems and do the same. It may not be the best answer, but it will still be safe and acceptable.
I'll discuss the rest of the paper tomorrow, but just really liked that opening sentence  so much that I just had to expand on it.
 The term "objectivity principle" is now being replaced by the "principle of material-frame indifference". In either case, the point is that the properties of a material should be the same regardless of the reference frame of the observer. In many simple models (i.e., ones lacking tensors), this is not the case. The classic example of this is using the general linear viscoelastic model for the case of a steady flow on a rotating turntable. The model shows that the viscosity depends on the rotational frequency, a result that is clearly wrong. See "Dynamics of Polymeric Liquids" by Bird, Armstrong and Hassager for details (Chapter 6). This is similar to equations that are dimensionally inconsistent.
 It's not a literary classic like "Call me Ishmael" or "It was a dark and stormy night", but for scientific writing, it pretty much ranks right up there.