One last note before jumping into it - I'd rather not talk about the inaccuracies in the new Cosmos. Tyson is not a historian of science which strongly suggests to me that he was handed a script for a cartoon narration. I don't think it has much of anything to do with the claims he makes here.
Second, my (again, under-educated) sense is that a lot of what goes by the name "philosophy of science" is actually epistemology and has little to do with actual science. I am thinking specifically of things like Popper saying that natural selection is not a scientific theory but (in his words) a "metaphysical research programme." At that point I am forced to simply say that that's nice as philosophical demarcation exercise, but at this point we've exited a discussion of science. This is not true of all philosophy of science by a long stretch (for all I know it's not even true of all of Popper - you'll have to ask someone that reads more Popper). Philosophers who spend more time thinking like philosophers as they inquire into the daily work of scientists - like Kuhn or Lakatos - generally avoid these excesses (since I'm trying to emphasize the extent to which I am not a philosopher, I'll note here that I have read Kuhn but not Lakatos, and very, very small bits of Popper). Why do I bring all this up? Because it can be frustrating for scientists to get told how to do things by "methodologists" or "philosophers of science" that are really just epistemologists who don't actually spend a lot of time thinking about the scientific enterprise.
Third, if you'll allow me to move forward with Kuhn, I see a tendency particularly in the social sciences, probably among young scientists of all types, and perhaps in the natural sciences to a certain extent too, an enormous yearning to do paradigm shifts rather than normal science. To say that someone is engaged in "normal science" is typically not a compliment. I think this is some of what Tyson might be getting at when he criticizes the focus on the "big questions". The vast majority of scientific effort is tedious, slow work on normal science. Paradigm shifts do not come around often and they are not something you often plan for either. Economics is rife with interest in what is essentially paradigm shift thinking and I think a lot of it is misguided. Mainstream economic science has its puzzles, but it is not in the sort of crisis that would foster a shift, and we certainly are not going to get a shift from legions of young economists going out in search of one. If a paradigm shift comes some day it's going to be a fairly singular event and while older ways of thinking about problems often resurface in paradigm shifts* the shift itself doesn't usually come from reading old books (which is what a lot of "big question"/"new thinking"/"dare to be different" types in economics seem to think). You don't do science by thinking about these big pie in the sky questions or ideas. If you really want to do science that generally means that you want to do normal science. 99% of the people who set out to be visionary paradigm shifters are deluding themselves. Tyson himself is a normal scientist. Most Nobel laureates are normal scientists and a lot of the ones that people would claim are paradigm shifters probably aren't. If you feel like doing normal science is beneath you you probably shouldn't work in science.
I don't know if I'd put the point the way Tyson did, but he was speaking off the cuff so I find it difficult to fault him too much for that. I do think underlying his comments are some very serious and reasonable points that people would do well to consider.
* - Kuhn does an excellent job illustrating this in talking about Aristotle, Newton, and Einstein on gravity where he points out that Einstein's perspective on the problem was quite Aristotelian in the sense that he went back to the Aristotelian view of gravity as having to do with inherent properties of objects - a view that Newton abandoned. I can't track it down, though - if anyone knows where that is in Structure of Scientific Revolutions I'd be very interested in you letting me know in the comments.