"For a chemist, concepts like aromaticity, acid-base behavior, functional groups and substituent effects wilt at the edges when they are defined too closely. They cannot be mathematicized, they cannot be defined unambiguously, yet they are of fantastic utility to our science".
'Wilt at the edges'- it's a lovely turn of phrase. What does it mean? It does not mean that there is no direct link between these concepts and their underlying explanation in terms of the physics of nuclei and electrons. There very much is. Instead to me it means something analogous to what another Nobel laureate, Steven Weinberg (quite a reductionist himself), said of the universe- "The more the universe seems comprehensible, the more it seems pointless". In case of chemistry it indicates that the closer we approach to purely physics-based explanations of chemical phenomena, the less useful and sensible those explanations seem to us as chemists. It's as if chemical understanding deserts us precisely when it seems to become satisfying for a physicist. That is probably part of what Hoffmann means when he refers to the fraying of chemical concepts at their edges.
To me the real problem with reductionism has not really been philosophical as much as practical. In some cases there is almost no tangible link between a phenomenon and its very deep underlying components. For instance, no one can reasonably draw a causative link between two people falling in or out of love or two countries going to war and the strong force between protons and neutrons. This is what we might term "strong reductionism" where the limitations of reductionist philosophy are stark and obvious. Philosophers love to discuss this kind. But I think that the type of reductionism that is far more relevant to the daily work of most scientists is what may be called "weak reductionism". This type applies to phenomena which can actually be connected to other basic phenomena in a reasonable sense, but it's just that explaining them in terms of these basic paradigms is utterly unhelpful at the level of the phenomena themselves.
Let me state an extreme example which would make this case clear. To a first approximation a physicist might say that there is no difference between a banana, a human being and a suspension bridge since all of them are made out of protons, neutrons and electrons. The only difference is really in their numbers. Almost everyone (including physicists) would realize that non-explanation as absurd. While it does seem like the ultimate unifying elucidation, it says absolutely nothing about the very different functions performed by a human, a banana and a bridge. All three of these respectable entities would resent their reduction to varying numbers of subatomic particles.
An example more familiar to chemists which is stated by Hoffmann also vividly illustrates the problem with reducing chemistry to physics. Consider the carbonyl functional group, a workhorse of chemistry. The most important reaction that this group undergoes is nucleophilic addition. How does physics explain this process? By essentially pitching electrostatics. Physics will tell us that the carbonyl carbon has a partial positive charge and the oxygen has a partial negative one, thus attracting nucleophiles to the carbon. But a chemist would find this simple explanation deeply unsatisfying. There is much complexity associated with addition to carbonyls which goes beyond merely electrostatic attraction. There's the angle of attack of the nucleophile- the well-known Burgi-Dunitz trajectory- which maximizes orbital overlap. There's coordination of positively charged counterions with the oxygen which can dictate the stereochemistry. There's also the size of groups on the attacking nucleophile which can sharply tip the distribution of products through steric effects. Then there's the gradation of reactivity of various nucleophiles based on their size and charge. And finally, there's the all-pervasive solvent which can drastically change product ratios and stereochemistry through solvation effects.
Now note that the truly fundamental underlying basis for all these factors (and indeed, virtually everything in daily life) is the electromagnetic force, and so yes, physics can purport to actually 'explain' all these factors by saying that they are all mediated through electromagnetism. Even steric effects are essentially electromagnetic in nature. Yet this would be akin to saying that wars happen because people get really angry at each other. The physics-based explanation is useless to the chemist since each one of the ingredients responsible for nucleophilic addition constitutes a unique chemical feature and conundrum which the chemist has to understand and predict. A unifying framework for these based on the physicist's conception of electromagnetism does nothing to delineate the special role that each factor plays in controlling the reaction. To a carbonyl-enamored chemist, these determinants are as fundamental in their own right as protons and neutrons are to a physicist.
Thus in such cases, reductionism fails not because there is no palpable connection between the chemical phenomenon and its physical underpinnings, but because the physics-based explanations tend to be useless at the level of chemistry. One of the simple tests for interrogating the utility of reductionism then consists of asking whether a reductionist approach can help truly explain a certain phenomenon at the same level that the phenomenon is embedded in its parent discipline. Sometimes it helps, more often than not it doesn't. The underpinnings of the American Civil War and World War 2 are different. Chemistry is not physics.