Is it therefore fair to say that in all physics problems, it is acceptable to use that nature converges on a unique solution to solve problems? Can we prove that this is a valid strategy?
Alright, you have some muddle we need to demuddle.
First and foremost, doing textbook physics problems is not exploring the nature of nature. In no way do the artificially created mathematical problems that comprise the end of a mechanics or optics chapter reflect nature, the philosophy of science, or even the philosophy of physics. They're pedagogical artifacts, and more specifically, they are oversimplified, domain-specific models generally presented as mathematical diagrams (SEP) imbued with deductive mathematical methods. From a pedagogical perspective, one might essentially say that they are cognitive scaffolding targeting a specific proximal zone of development in a student. You've digested a bunch of pre-calc and calc, you have a highly abstracted, shortly written context, and you are picking out salient features in the semantics of physics to set up and solve some math to come to a conclusion. So, that convergence you note on a single way that results in a unique solution for the problem? Yeah, that's the textbook author's design, and not a facet of nature. In fact, one of the seminal and controversial ideas between philosophers of science, is one of the question of underdetermination. Radical instrumentalists will insist that no amount of explication will ever prove a theory adequately, though most philosophers take a heavily realist approach, and believe in positions that advocate, like some generalized version of Cummins's functional analysis, that additional iterations of empirical process will resolve any ambiguity that inheres regarding how data maps to justified theory. You state the third step in problem solving is:
Verify that the solution confirms the postulate.
In math class, sure. But in science, the notion of verification (and confirmation) have been abandoned by philosophy of science thinkers, though obviously not in practice; Karl Popper is famous for his rejection of verificationism and confirmationism with his proposal of falsifiability. In a textbook, you verify an answer when you model the problem correctly (often using derived formulas that are part of the canonical theory) by calculating, checking your math, and then checking the solution manual. Again, this activity isn't Nature. It's the publisher, and a means for an evidenced-based evaluation of a students performance for grading, much in the same way researchers use shallow citation metrics to measure their impact or contributions to science (which is to say, largely a useless cultural artifact used for political reasons, with the exception of rare papers that are tremendously influential) to keep their jobs in publish or perish environments. The problem solving you're focusing on might be seen as perpetuating normal science (SEP), under Kuhn's theory put forth in The Structure of Scientific Revolution.
The practice of physics is much broader than textbook problems, and if you're interested in what physics is more broadly, at least through the lens of an empiricist who prides himself on creating and testing variables, take a look at Martin Krieger's Doing Physics (GB). In great detail, he explicates how the scientific apparatus is designed to create the math necessary to isolate and prove scientific hypotheses. One you get beyond textbook problems aren't really physics in the sense of an empirical methodology, then you're ready to begin exploring the philosophy of physics and the philosophy of science more broadly. For the former, I'd recommend *Philosophy of Physics: A Very Short * (GB), and for the latter I'd recommend A Companion to the Philosophy of Science (GB). I poked around the SEP and found Experiment in Physics, which while not my cup of tea, does looked like a solid introduction into the empirical nature of physics. Exercises for the classical mechanics, E&M, and modern physics (usually taught as a three-course, calculus based sequence) are just to help you develop the bare bones necessary for understanding what physics is.
(PS And Nature, Natural Kinds, and the Laws of Nature have largely been deprecated as concepts, but that's another Q&A response entirely.)