I'm curious to how it could have been improved by insights from academic philosophy. I'm not expert but glanced at the Wikipedia entry on philosophy of science and find it hard to see what you'd go with.
Judea Pearl - Causality
John Earman - World Enough and Space-Time: Absolute vs. Relational Theories of Space and Time
James Woodward - Making Things Happen
Jeff Bub - Interpreting the Quantum World
Peter Godfrey-Smith - Darwinian Populations and Natural Selection
Jesse Prinz - The Conscious Brain
Kim Sterelny - Thought in a Hostile World
And some historically important works:
Carnap - The Logical Structure of the World (in a loose sense, the first attempt at an AI program)
Popper - The Logic of Scientific Discovery (the philosophy of science practicing scientists now inherit and unquestionably assume)
Kuhn - The Structure of Scientific Revolutions (a ubiquitous work on scientific practice)
Readers of HN might also dig up Scott Aaronson's paper on how complexity theory might be applied to certain philosophical questions.
Just to give you a taste of what to expect though: imagine a lot of very smart people considering the sorts of issues Feynman is raising, but devoting whole books to them, trying to cover these issues from every conceivable angle and taking in to account all of the other points of view on these issues (not just those of Feynman himself or some astrologers or psychics, as Feynman did). Then there are the books in response to those books, trying to be just as thorough in answering the questions the the first author wrote about, and seeing if there's something they missed or any other constructive way to think about them, and so on.
In short, there's an entire academic field that studies these questions and many other related ones, that isn't going to be covered by a little essay by any single scientist -- no matter how brilliant he was. He didn't have the final word, and not even much of a word in the first place, considering his lack of study of and engagement with the field.
Feynmann's own words in the essay on "Cargo Cult Science" are salient here:
"It’s a kind of scientific integrity, a principle of scientific thought that corresponds to a kind of utter honesty--a kind of leaning over backwards. For example, if you’re doing an experiment, you should report everything that you think might make it invalid--not only what you think is right about it"
It's telling that Feynmann does not apply this principle of integrity to the views he expresses in his own essay. He does not, even for a moment, write about what could possibly be wrong with his point of view. But such questioning is quite common in the philosophy of science. They don't stop with dogmas and ideals regarding what science ought to be, as Feynman does. They go further and question them, and much else besides.
By "do better" I mean in an equivalent few pages to advise a new graduate about how to discover things that are true, as a focus sometimes at odds with advancing their career, because that's what Feynman's graduation speech was about.
Philosophy of science is less about telling scientists how to behave than trying to understand what actually happens when scientists do what they do, what science accomplishes on a theoretical level, what the difference between science and non-science is, etc.
It's less of "here's how to get to the truth" and more of "can science get at the truth at all?" or "to what degree can science get at the truth?" or (much like Feynman's essay) "what makes these particular actions or beliefs 'scientific'?"
Karl Popper's criterion of "falsifiability"[1] was once a popular one, in regards to what makes something 'scientific'. But that's fallen out of fashion[2]:
"Sir Karl Popper is not really a participant in the contemporary professional philosophical dialogue; quite the contrary, he has ruined that dialogue. If he is on the right track, then the majority of professional philosophers the world over have wasted or are wasting their intellectual careers. The gulf between Popper's way of doing philosophy and that of the bulk of contemporary professional philosophers is as great as that between astronomy and astrology."
And so the debate continues...
[1] - https://en.wikipedia.org/wiki/Falsifiability
[2] - https://en.wikipedia.org/wiki/Falsifiability#Criticisms
That doesn't really mean Popper is wrong.
If it can't then by its own measure, it's not "scientific". Further, if one scorns anything "unscientific" as valueless, then the principle of falsifiability must also be valueless.
In the relevant sense, yes. We switch out conceptual definitions when they fail to encompass the relevant empirical instances. If falsifiability turned out to allow, for example, theology to be considered a science, then we would have to throw out falsifiability. As it is, it's only a coarse-grained principle for demarcation, hence the arguments about string theory.
I think that may have been Feynman's point.
Whether, or to what extent, Popper's falsifiability criterion is a good description of science is not a problem for scientists.
On the "why questions" video from the first link, which I've seen before, Feynman's answer only shows that he's used to answering the kind of question asked of him, regarding why magnets repel, and similar questions involving physics.
Yes, he realizes that there are larger issues involved in such questions and their answers, but he steers all the answers and questions back to physics, which to him are the more interesting answers and questions.
Nowhere is there any evidence that his own answers are informed by any kind of reading of philosophical literature or familiarity with the world of philosophy.
Regarding the "Cargo Cult Science" essay. In it Feynman states "even today I meet lots of people who sooner or later get me into a conversation about UFO’s, or astrology, or some form of mysticism, expanded consciousness, new types of awareness, ESP, and so forth. And I’ve concluded that it’s not a scientific world."
His interest in the question of what is and isn't science, and his answer to it are some of the central things that philosophy of science studies. No amount of physics experiments are going to give you any insight in to that question. To attempt to answer it, as Feynman did, is to practice philosophy of science, which Feynman himself in the original quote which started this thread decries as useless.
The whole essay is really an attempt to engage in philosophy of science, but without any evidence that Feynman is familiar with the field. The result is interesting both because Feynman is an entertaining and capable writer, and because of his obvious familiarity with the practice of science and its ideals. But it's not nearly as insightful as it could have been had he actually undertaken to study the subject academically. It's really an opinion piece aimed at explaining in very general ways some of the practices and ideals of science as Feynman sees them to laymen and other scientists, and almost a polemic on scientific integrity.
Also, it should be pointed out that the philosophy of science studies a lot more than just the questions addressed in this essay. They are important questions, but there's a lot more about science that is philosophically interesting.
Regarding your point about the clarity of Feynman's writing compared to those of philosophers: it's sadly all too true that way too many of them are very poor and obscure writers compared to Feynman. On the other hand, many others are very clear (if usually pretty dry).
When there is lack of clarity, some of it is due to lack of writing ability or interest in clarity (or even wilful obscurity), but in other cases it's due to the specialized knowledge or terminology that is required to understand it -- much like that which would be required to understand a paper in an academic physics journal. If you attempt to tackle such a paper without a great deal of academic physics under your belt, you are likely to be confused. So it is with philosophy, where you'd be a lot better off with some philosophy courses under your belt before you try to make sense of much of it.