I always enjoy your essays, Sabine. Please keep writing! This one inspired me to enter this contest too.
In your signature style, you continue to emphasize two points physicists need to be continually reminded of: physics is not math, and experiment is the ultimate arbiter of truth in physics.
Some great turns of phrase I liked:
"If it's not a deadline that sets an end to your hesitation, then the heat death of the universe certainly will."
"They have never been deterred by not knowing whether what they aspire to is even possible, and hopefully they never will."
"And, looking at the literature on black hole collapse, I fear we may not answer this question in finite time either."
Excellent point about mathematical problems crucially dependent upon some kind of infinity not being very relevant to science. Another essay I read here (by Michael Kewming) made the similar point that a computer trying to solve a halting problem would halt eventually, due to physical limitations, the omnipresence of noise, its eventual degradation, etc.
Probably the one place where I have any substantive disagreement with you is about chaos, or unpredictability, or whatever you want to call it. In particular, I'm not sure I agree that linear = predictable. Maybe in a limited sense, e.g. if Laplace's demon knows the wave function of the universe at one time, and can perfectly solve the Schrodinger equation, then linearity means small inaccuracies in its initial knowledge do not blow up out of control. But in practice, we can only learn about the universe by making measurements of observables. Taking those measurements can affect the outcomes of future measurements in unexpected ways (see some of the recent literature on out-of-time-ordered correlators...which you may already be familiar with). In this sense, I think unpredictability is also a feature of quantum mechanical theories.
Very beautiful point about unpredictability in phenomenological theories being important, and that it can signal that interesting things are happening (e.g. hot plasma instabilities). Makes me think of how all kinds of singularities appear in physics (like particle bumps), but we never actually measure an infinitely large bump. The singularity is just a useful approximation.