"solves differential equations" doesn't tell you anything about which differential equations. "minimizes action" gives you a way to get the differential equations corresponding to pretty much every scenario.

Specifying the action is equivalent to specifying the differential equations. Knowing that you start with a Lagrangian doesn't tell you more than knowing you start with some differential equations.

But are the statements "reality minimizes action" and "reality solves differential equations" actually, literally equivalent to each other? It strikes me that this is not in fact the case. There are solutions to systems describing skyscraper sway that have the building stand at a terrifying lean angle; what is supposed to be selecting the particular solutions that we observe? A purely random walk through the solution space seems unsatisfying as an answer.

They are actually, literally equivalent to each other. Differential equations aren't a "purely random walk," they are a statement of what is about to happen based on what is happening. You have three formalisms: Newtonian (differential equations), Lagrangian (action minimization), and Hamiltonian (differential equations, but derived from a scalar field kind of like in Lagrangian mechanics). They are all different ways of writing down the same thing, and their advantages and disadvantages are related to situational convenience.

Thank you for the response, I find this very interesting.

So is it true to say that the production of a particular solution to a differential equation is a statement about how a system will behave based on its initial conditions, and that the statement captures within it the principle of action minimization by virtue of the fact that it is a derivation of information from natural laws?

"Action is minimized" and "F = ma" are the same in the sense that x+5=0 and x+6=1 are the same. In both cases you can go from one to the other with a sequence of deductions. There's no way to tell which one is the "natural law" because you can take either one as a fact and derive the other as a consequence.

Interesting - your response makes sense. Thanks again for the information.

I just assume physicists are weirdos like that, they want reality to do easy and nice things, not math :)

As you see it clicks for many people, and doesn't mean much for others. Back then it was a very welcome click.

After all, it should be possible to find antiderivatives and solve ODEs without doing the dance with substitutions and change of variables, but still, we find it many magnitudes easier than just looking at it and divining the correct solution.