These three questions are phrased as alternative-history questions, but my real intent is to understand better how well different modeling approaches fit the phenomena they are used to describe; see 1 below for more discussion of that point.
Short "informed opinion" answers are fine (better, actually).
If Dmitri Mendeleev had had access to and a full understanding of modern group theory, could have plausibly structured the periodic table of chemistry in terms of group theory, as opposed to the simpler data-driven tabular format that he actually used?
If Mendeleev really had created a group-theory-based Periodic Table, would it have provided any specific insights, e.g. perhaps early insights into quantum theory?
The inverse question: If Murray Gell-Mann and others had not used group theory concepts such as $SU(3)$ to organize particles into families, and had instead relied on simple grouping and graphical organization methods more akin to those of Mendeleev, is there any significant chance they could have succeeded? Or less speculatively, is it possible to create useful, concise, and accurate organizational structures (presumably quark based) that fully explain the particle data of the 1970s without making any reference to algebraic structures?
1 Background: My perspective on the above questions is to understand the interplay between expressive power and noise in real theory structures. One way to explain that is to note that mathematical modeling of data sets has certain strong (and deep) similarities to the concept of data compression.
On the positive side, a good theory and a good compression both manage to express all of the known data by using only a much smaller number of formula (characters). On the negative side, even very good compressions can go a astray by adding "artifacts," that is, details that are not in the original data, and which therefore constitute a form of noise. Similarly, theories can also add "artifacts" or details not in the original data set.
The table-style periodic table and $SU(3)$ represent two extremes of representation style. The table format of the periodic table would seem to have low expressive power and low precision, whereas $SU(3)$ has high representational power and precision. The asymmetric and ultimately misleading emphasis on strangeness in the original Eight-Fold Way is an explicit example of an artifact introduced by that higher power. We now know that strangeness is just a fragment -- the first "down quark" parallel -- of the three-generations issue, and that strangeness showed up first only because it was more easily accessible by the particle accelerators of that time.
2012-06-30 - Update on final answers
I have selected Luboš Motl's answer as the most persuasive with respect to the questions I asked. If you look at the link he includes, you will see that he has looked into this issue in minute detail with regards to what kind of representation works best, and why. Since that issue of "what is the most apt form of representation" was at the heart of my question, his answer is impressive.
With that said, I would also recommend that anyone interested in how and to what degree group theory can be applied to interesting and unexpected complexity problems, even if only approximately, should also look closely at David Bar Moshe's fascinating answer about an entire conference that looked at whether group theory could be meaningfully applied to the chemical elements. This excellent answer points out a rich and unexpected set of historical explorations of the question. If I could, I would also flag this as an answer from a different perspective.
Finally, Arnold Neumaier's answer shows how a carefully defined subset of the problem can be tractable to group theoretic methods in away that is predictive -- which to me is the single most fundamental criterion for when a model crosses over from being "just data" into becoming true theory. And again, I would flag this one as an answer if I could.
Impressive insights all, and my thanks to all three of you for providing such interesting, unexpected, and deeply insightful answers!