In my opinion the article mixes up three distinct notions and fuses them into one to create the impression that physics is less solid than it actually is.
These would be:
1.
Undergraduate inconsistencies. A few times in the articles he demonstrates how putting together certain arguments from different courses in undergraduate physics one can derive contradictions. This is to be expected, since undergraduate physics mixes different topics at different levels. For example typically we teach relativity to a much deeper level than statistical mechanics, so I've had several undergraduate students ask me about apparent contradictions between Stat. Mech. and relativity, which only result from an imbalance in the amount they know of both subjects.
Similarly the example of the "infinite energy" in the gravitational potential. As nwr already said this only arises because one uses explicitly unphysical matter, matter which the Newtonian theory does forbid. Of course I recognise that the author knows this, but I don't think it's a good example of "The man behind the curtain", because "behind the curtain" is the real explanation: use only physical matter with Newton's equations.
2.
Technical issues resolved decades ago This includes examples such as renormalisation, which is my specialty. Again there is no "man behind the curtain", renormalisation was originally a confusing and highly technical subject that slowly became clearer between 1960 - 1980. Although it's one of my favorite areas, I have to admit it's largely just a "boring" technical problem relating to how exactly you define a quantum field theory mathematically, it doesn't really reflect gaps in our physical understanding.
3.
Real issues There is only really one of these in the whole article, which is the issue of what quantum mechanics is really all about. This definitely hasn't been resolved.
I think the author should have written more about the real issues, instead of taking examples from all over the place (particularly undergraduate teaching) to give examples of contradictions or holes in physics.
I certainly don't agree with the author's main conclusion, as I don't see any major cracks in the edifice. There are unsolved problems sure, but none of the established stuff is self-contradictory. I would say the only real major "man behind the curtain" at the moment, where I take this phrase to mean an abundance of unresolved issues in an area we say is a closed book, would be the transition between quantum mechanics and molecular physics.
We know atoms and particles are described well by quantum mechanics. For molecules, particularly crystal molecular structures like salt, we use (probabilistic) models that you can justify in a hand-waving way, but have never really been shown to actually follow from quantum mechanics, however I don't view this as a major problem.
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