Their point in SICM was that traditional notation was not as ambiguous as their computational notation. That's because traditional notation is designed to be convenient rather than explicit. They are right, but this does not mean that physics lacks rigour. More explicit notations were always available and physicists, being more intelligent than computers, were capable of resolving the ambiguities to reveal the fully rigorous structure underneath.
There's a difference between using a convenient notation and lacking rigour. If you are always capable of discerning the true formal equations behind a convenient notation then everything is OK. If you can't, or people can't agree on what the formal meaning should be, then there is indeed a problem. The fact that the authors could derive a computational notation that no physicist would disagree with is proof that a lack of rigour never existed in the first place.
You write that "More explicit notations were always available and physicists, being more intelligent than computers, were capable of resolving the ambiguities to reveal the fully rigorous structure underneath."
I doubt that's true. For example, here's what Piet Hut, now a professor of physics at the Institute for Advanced Studies at Princeton, writes about his experiences with classical mechanics as an undergraduate in his review of SICM available at
http://www.ids.ias.edu/~piet/publ/other/sicm.html :
"Soon I went through the library in search of books on the variational principle in classical mechanics. I found several heavy tomes, borrowed them all, and started on the one that looked most attractive. Alas, it didn't take long for me to realize that there was quite a bit of hand-waving involved. There was no clear definition of the procedure used for computing path integrals, let alone for the operations of differentiating them in various ways, by using partial derivatives and/or using an ordinary derivative along a particular path. And when and why the end points of the various paths had to be considered fixed or open to variation also was unclear, contributing to the overall confusion.
Working through the canned exercises was not very difficult, and from an instrumental point of view, my book was quite clear, as long as the reader would stick to simple examples. But the ambiguity of the presentation frustrated me, and I started scanning through other, even more detailed books. Alas, nowhere did I find the clarity that I desired, and after a few months I simply gave up. Like generations of students before me, I reluctantly accepted the dictum that `you should not try to understand quantum mechanics, since that will lead you astray for doing physics', and going even further, I also gave up trying to really understand classical mechanics! Psychological defense mechanisms turned my bitter sense of disappointment into a dull sense of disenchantment."
I actually agree with Sussman and Wisdom (and the author of this article) that physics is taught in a very sloppy manner, and that the computational notation has huge advantages. But that's not the point I (or they) were making. Lagrangian and Hamiltonian mechanics were perfectly rigorous before Sussman and Wisdom came along, even if they weren't taught very well. If mechanics had really been deep-down sloppy, then Sussman and Wisdom would now be heralded alongside Newton and Einstein for their incredible contribution to science! Rather than just creating a new notation, they would have advanced science incalculably, turning vague and untestable theories into hard empirical science for the first time!
But that's obviously not what happened. Physicists were always capable of making exact calculations and predictions from Lagrangian mechanics. SICM contains no new theories, theorems or proofs, just a more explicit way of representing old ones. Nothing new was discovered and no old notions were clarified. Instead, they just found a better way of teaching mechanics, one that didn't rely on the implicit knowledge that masters of the subject already possessed. They were only capable of doing this because Lagrangian and Hamiltonian mechanics were well-defined in the first place. If they hadn't been then they would have had to advance a new theory of mechanics to replace them, rather than just re-presenting an old one.
There's a difference between using a convenient notation and lacking rigour. If you are always capable of discerning the true formal equations behind a convenient notation then everything is OK. If you can't, or people can't agree on what the formal meaning should be, then there is indeed a problem. The fact that the authors could derive a computational notation that no physicist would disagree with is proof that a lack of rigour never existed in the first place.