I think your question here can be decomposed into several different questions. When asking why people choose something, you have to compare that choice to the alternatives. You posit that the alternatives to physics are math and computer science. I don't see that. The more common alternatives for physics are biology and chemistry. In highschool I have to take one science course each year and one math course each year. I could choose between physics, chemistry, and biology. Those were the options. If someone grows up thinking they want to be a scientist, they major disciplines of science are typically considered physics, chemistry, biology, medicine. Why physics as opposed to biology is a very different question than why physics as opposed to math.
"Why physics?" is a question people will tend to have their own answers to, but I've watched a fair number of undergraduates, and a lot of the time it's possible to identify the ones that will become physics majors, at least as opposed to majoring in chemistry/biology/premed. It's harder to pick between physics majors and math majors. Lots of physics majors double major in math. Many computational physicists have degree in physics and computer science. Picking which students will becomes physicists versus chemists or doctors is relatively easy though, and it's not a matter of intelligence. It is about attitude. Part of the physics attitude is summed up in the Feynman quote, "I learned very early the difference between knowing the name of something and knowing something."
In most subjects, especially in biology and chemistry, the names of things are very important. A lot of work goes into learning the names of things, and those names stay pretty consistent. In physics that isn't true. The letter V may be voltage in this problem, or velocity, or mass. It doesn't matter. There are lots of students who if I write that a block 1 has a mass V1=5 kg will absolutely refuse to go any further. They can't handle that. To me, and to many of the people that go on to study physics, that was a breath of fresh air. Finally, we're learning ABOUT something, rather than just learning the name of something.
Physics allows you to make predictions about the world. How fast does something fall? Which parts of this socket can I safely stick my fingers into? Math doesn't give you that. Computer science only allows you to predict how human created systems work. Physics allows me to understand the world, rigorously. I still remember learning that starfish are in phylum echinodermata. Maybe that's a useful fact, but it doesn't feel nearly as good as understanding that the frequency of a pendulum has to be sqrt(g/l) and not sqrt(l/g), because that's the only way the units make sense.
I think you're also underrating how useful studying physics is. Math is in general pretty difficult for me to learn. It's abstract, and it isn't always obvious to me where the rules are coming from. Almost all of the math I know more complicated than calculus I learned in physics classes. Even if I took a math class on a subject, it typically wasn't until i covered in it a physics class that I really understood the concept. I may not use very many of the results of quantum mechanics on a day to day basis, but QM gave me a firm grounding in probability theory and in abstract vector spaces, and those are both useful concepts. I would probably disagree with your claim that thermodynamics (which I consider a chemistry course) is more useful than the more physics-oriented equivalent: statistical mechanics, since the latter covers not only heat engines, but also is a good introduction to both statistics and information theory. As a result of my physics education I can look at almost any object or machine and I have a pretty good guess as to how it works, from maglev trains to GPS satellites to MRI machines. That's a pretty useful skill. I'm also really good at estimating the order of magnitude of numbers. That comes in handy too.
I don't know anything about studying physics to signal intelligence, or studying physics because that's where the smart people were. Until my social circle became predominantly comprised of physicists, the smartest people I knew were never physicists, and were never that impressed that I was studying physics. People who don't consider themselves highly-intelligent who I meet in casual conversation do occasionally make the comment, "You study physics, you must be really smart." but saying I look at models of phenomenon X in department Y, trying to predict Z is probably enough to get you branded as "really-smart" by most people.
Your P.S. is completely unsuprising. Premeds are in general terrible physics students. They're very clever at patten matching and faking it, but you can trip them up with the most minor disruptions to the pattern, like using h as the variable for velocity instead of v. They're like neural networks really. They interpolate really well, but they couldn't extrapolate themselves out of an open paper bag.
No one pushed me into physics. No one suggested I study it. I studied history, philosophy, evolutionary biology, theology. I have an undergraduate degree in English, which a focus on poetry writing. But my graduate education is primarily in physics, with a hint of computer science. I'm haven't been exposed to a strong social pressure towards physics, and I haven't seen anything that would lead me to believe that people overvalue physics.
Math and computer science aren't regarded as science. There's no Bill Nye the Math Guy. Smart kids want to be scientists. Once you've decided to be a scientist, physics is a pretty natural choice.
Many of the high school students who sought advice from Cognito Mentoring were interested in mathematics, computer science, and physics. This both makes sense and is valuable. Mathematics has many benefits: it underpins a lot of quantitative analysis, and helps us understand the world. Computer science is also quite important for obvious reasons: programming in particular is directly and indirectly useful, and a deeper understanding of algorithms and the theory of computation can help with algorithms.
Physics, however, is a little different. There are some benefits of learning physics. In particular, classical mechanics is often people's first exposure to using mathematical structure in a nontrivial way to understand and model situations pertaining to the real world. Nonetheless, unlike mathematics or computer science, the benefits of physics for people who are not in science or engineering careers are fairly low. I find myself using high school-level mathematical intuition on a regular basis (for instance, understanding the growth trajectories of various things, or interpreting graphs), and I find myself using basic programming-like intuition quite often. But I rarely find myself using my physics intuition in the real world. Moreover, I think physics quickly hits diminishing returns in terms of teaching people about mathematical modeling: I'd say that the returns from physics beyond classical mechanics, DC circuits, and basic thermodynamics are near-zero. For instance, I'd say it's more beneficial to learn microeconomics rather than electromagnetism, even though the latter is often considered more prestigious by smart people. Similarly, I think that behavioral economics is more valuable than quantum mechanics.
It's also not clear that learning physics beyond the basics suggested above (classical mechanics, thermodynamics, DC circuits) passes a cost-benefit analysis for people in the vast majority of science-based and engineering-based careers. Even the extent to which they crucially rely on these basics is questionable, given that most people don't learn the basics well and still manage to go on to do decent jobs. I'd like to hear any opinions on this. On a related note, I recently asked on Quora the question In what ways is knowledge of Newtonian classical mechanics helpful to people pursuing biomedical research? and there were a few interesting answers.
So my question: what attracts smart and curious young people to physics? Are the smartest people too attracted by physics, relative to its real-world applicability? Does the intellectual stimulation provided by physics justify the attraction? Is there some sort of mood affiliation going on here, where the smartest people are pulled to physics to distinguish themselves from the crowd, insofar as physics is more difficult and repels the crowd? To the extent that people overvalue physics, does it make sense to push them at the margin away from physics and in the direction of computer science or economics or some other subject? Or should their interest in physics be encouraged?
Thoughts on your personal experience, as well as thoughts on the general points about the usefulness and attractiveness of learning physics, would be appreciated.
PS: In a video, Eric Mazur describes research related to the Force Concept Inventory: people often learn how to solve complicated mechanics problems by pattern-matching but fail to demonstrate clear understanding of Newton's Third Law. Similarly, people can predict potential differences and current flows in complicated circuits using Kirchhoff's laws, yet fail to predict that if you short a circuit, all the current will flow through the short. (The latter failure of prediction occurred in an end-of-course examination co-taught by Mazur to Harvard University first-year students, many of whom were planning to go on to medical school.
PS2: My collaborator Jonah Sinick's Quora post (no login needed to view) titled Is math privileged for gifted children is somewhat related.